Fluxon Dynamics of the 1j0 State in a Two-Fold Stack of ...hadisusanto.net/paper2/MSc_HA_Kooiker.pdf · iv This report is written in MIKTEX 2.1 with WinEdt v. 5.3. Most important

Fluxon Dynamics of the [1|0] State in aTwo-Fold Stack of Long Josephson Junctions

H.A. Kooiker(s9706720)

Graduate thesis for the department “Applied Mathematics” (chair “AppliedAnalysis & Mathematical Physics”) of the University of Twente.Examination committee: Prof. dr. S.A. van Gils,

H. Susanto MSc.,Faculty of Electric Engineering, Mathematics andComputer Science.Dr. A.A. Golubov,Faculty of Science and Technology.University of Twente, Enschede (The Netherlands).

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iii

Fluxon Dynamics of the [1|0] State in aTwo-Fold Stack of Long Josephson

Junctions


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iv

This report is written in MIKTEX 2.1 with WinEdt v. 5.3. Most important book of referenceis “A guide to LATEX2ε”1 by Helmut Kopka and Patrick W. Daly.

1H. Kopka and P.W. Daly. A guide to LATEX2ε, Document Preparation for Beginners and Advanced Users.ISBN 0-201-42777-X. Addison-Wesley Publishers Ltd, Edinburgh Gate, Harlow, Essex, second edition (1995),1993.

June 2005 Fluxon Dynamics of the [1|0] State in aTwo-Fold Stack of Long Josephson Junctions

v

Abstract

Since the discovery of superconductivity in 1911 by the (for his discovery with a Nobel Prizeawarded) Dutch physicist Heike Kamerlingh Onnes, great efforts have been devoted to findingout how and why the phenomenon of superconductivity works. During the 1950s, theoreticalcondensed matter physicists arrived at a solid understanding of “conventional” superconduc-tivity, which is (among other things) captured by the important BCS theory, developed byBardeen, Cooper & Schieffer in 1957. Originating from this theory, combined with quantummechanical knowledge, in 1962 Brian Josephson predicted and also showed the existence ofthe Josephson effect, i.e. the possibility of a current to tunnel through a thin insulator fromone superconductor to another. This configuration of two superconductors separated by athin insulator is known as a Josephson junction. An important consequence of the Josephsoneffect is the possibility of a fluxon to exists in a Josephson junction. This fluxon is a circu-lating current across the insulator due to the phase difference between the electron’s wavefunctions in the superconductors. This fluxon can be forced to move along the junction byapplying an exterior bias current to the junction’s superconductors. Nowadays, the Joseph-son junctions are exploited for various applications like, e.g., extremely sensitive measurementtools (SQUIDs), digital circuit operations and as radiation source.

The phase difference between the electron’s wave functions in the superconductors of a Joseph-son junction can be modelled by a perturbed variant of the sine-Gordon equation. Hence,the system of equations that govern the phase differences over the Josephson junctions in astack is given by a set of coupled perturbed sine-Gordon equations. These equations admit2π-kink travelling wave solutions, which correspond to a fluxon in the Josephson junction.By applying regular perturbation theory, analytic expressions for the phase differences overthe junctions in the stack can be derived. These expressions for the phase differences in astack of two coupled junctions are determined up to order ε (the perturbation parameter),where there is a fluxon considered to be present in the one of the junctions and no fluxon inthe other (the [1|0] state). These expressions require an unique relation involving the velocityof the fluxon, the applied bias current to the stack and the damping of the insulators to besatisfied, in order for the fluxon to exist for small perturbations of the system.

The stability of travelling wave solutions to the coupled perturbed sine-Gordon equations canbe determined by investigating the essential spectrum and the point spectrum. The essentialspectrum determines the time behaviour of oscillating perturbations to the travelling wavesolutions. In particular, the boundary of the essential spectrum determines the behaviour overtime of oscillating perturbations to the asymptotic states of the travelling wave solutions. Thepoint spectrum, which is disjunct of the essential spectrum, determines the behaviour overtime of bounded (non-oscillating) perturbations. The (boundary of the) essential spectrum ofthe travelling wave solutions, corresponding to the [1|0] state of the stack, is shown to residein the left half plane and hence, points in the essential spectrum have non-positive real part.


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vi Abstract

Also the point spectrum is shown to reside in the left half plane, although this could only beshown for junctions in the stack which are identical. By this it is proven that the travellingwave solutions for small enough perturbations to the system are linearly stable solutions ifthe junctions in the stack are identical.

By using the software package AUTO, the parameter branches for which there exist travellingwave solutions to the coupled perturbed sine-Gordon equations are computed, where theperturbations do not necessarily need to be small. It has been observed that the dependenceof the velocity of the solutions on the bias current is determined by the ratio of the criticalcurrents of the junctions. Therefore, the situations that this ratio is larger than, equal to andsmaller than 1 are distinguished. For all three situations, spirals are observed in the parameterspace, where there is numerical evidence that for the parameter values at the centres of thespirals heteroclinic solutions exists, different from a fluxon which corresponds to a homoclinicsolution. Furthermore, the transition of the found solution branches is computed if the ratioof the critical currents goes from larger than 1, through 1, to smaller than 1.


vii

Samenvatting

Vanaf het moment dat superconductiviteit werd ontdekt in 1911 door de nederlandse natu-urkundige Heike Kamerlingh Onnes (die hier later de Nobel Prijs voor kreeg) zijn er veelpogingen gewijd aan het uitzoeken hoe en waarom superconductiviteit werkt. Gedurende dejaren ’50 kwamen theoretische natuurkundigen tot een solide basis van de “conventionele” su-perconductiviteit, die (onder andere) is beschreven in de belangrijke BCS theorie, ontwikkelddoor Bardeen, Cooper & Schieffer in 1957. Voortkomend uit deze theorie, in combinatiemet kwantum mechanische kennis, voorspelde in 1962 Brian Josephson het bestaan van hetJosephson effect, waarvan hij het bestaan ook aantoonde. Het Josephson effect is de mogelijkdat stroom door een dunne isolator van een superconductor naar een andere kan lopen. Dezeconfiguratie van twee superconductoren, gescheiden door een dunne isolerende laag, staatbekend als een Josephson junctie. Een belangrijk gevolg van het Josephson effect is het mo-gelijk maken van het bestaan van een fluxon in een Josephson junctie. Deze fluxon is eencirculerende stroom door de isolator als gevolg van het fase verschil tussen de golfvergelijkin-gen van de electronen in de superconductoren, die langs de junctie kan worden bewogen dooreen externe stroom door de superconductoren te sturen. Tegenwoordig worden Josephsonjuncties gebruikt in verschillende toepassingen zoals, bijvoorbeeld extreem gevoelig meetap-paratuur (SQUIDs), operaties van digitale circuits en als stralingsbron.

Het fase verschil tussen de golfvergelijkingen van de electronen in de superconductoren van eenJosephson junctie kan gemodelleerd worden door een verstoorde variant van de sinus-Gordonvergelijking. Hierdoor is het systeem van vergelijkingen, die de fase verschillen van meerdereJosephson juncties in een stapel beschrijven, gegeven door een verzameling van gekoppeldeverstoorde sinus-Gordon vergelijkingen. Deze vergelijkingen kunnen 2π-kink lopende golvenals oplossing hebben, die corresponderen met een fluxon in een Josephson junctie. Doorreguliere storingstheorie toe te passen kunnen de analytische uitdrukkingen voor de fase ver-schillen in gekoppelde Josephson juncties bepaald worden. Deze uitdrukkingen voor de faseverschillen van twee gekoppelde juncties zijn bepaald, tot aan uitdrukkingen van order ε (destoringsparameter), waar er verondersteld is dat een fluxon zich bevindt in een van de junctiesen geen in de andere junctie (de [1|0] staat). Deze uitdrukkingen vereisen van de snelheid vande fluxon, de toegepaste externe stroom en de remmende werking van de isolator dat ze aaneen unieke vergelijking voldoen, zodat de fluxon kan bestaan voor kleine verstoringen aan hetsysteem.

De stabiliteit van de lopende golfoplossingen van de gekoppelde verstoorde sinus-Gordonvergelijkingen kan worden bepaald door het essentiele spectrum en het punt spectrum teonderzoeken. Het essentiele spectrum bepaalt het tijdsgedrag van oscillerende verstoringenvan de lopende golfoplossingen. In het bijzonder bepaalt de rand van het essentiele spectrumhet tijdsgedrag van oscillerende verstoringen van de asymptotische staat van de lopende gol-foplossingen. Het punt spectrum, wat disjunct is van het essentiele spectrum, bepaalt het


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viii Samenvatting

tijdsgedrag van begrensde (niet oscillerende) verstoringen. Voor het geval van de [1|0] staatin de gekoppelde juncties is aangetoond dat (de rand van) het essentiele spectrum zich in hetlinker halfvlak bevindt en dus hebben punten in het essentiele spectrum een niet-positief reeeldeel. Ook voor het punt spectrum is aangetoond dat dat zich bevindt in het linker halfvlak,alhoewel dit alleen aangetoond is voor twee juncties die helemaal identiek zijn. Hiermee isbewezen dat de lopende golfoplossingen van het verstoorde gekoppelde systeem lineair stabielzijn, zolang de verstoringen klein genoeg en twee juncties identiek zijn.

Met behulp van het software pakket AUTO zijn de parametercurves berekend, waarvoor erlopende golfoplossingen bestaan van de gekoppelde verstoorde sinus-Gordon vergelijkingen,ook als de verstoringen niet noodzakelijk klein zijn. Er is gezien dat de relatie tussen desnelheid van de oplossingen en de externe stroom wordt bepaald door de ratio van de kritiekestromen van de juncties. Daarom is er onderscheid gemaakt tussen de situaties dat deze ratiogroter dan, gelijk aan en kleiner dan 1 is. Voor alle drie situaties zijn er spiralen gevonden inde parameterruimte en er is numeriek bewijs dat voor de parameterwaarden in het centrumvan de spiralen er heterokliene oplossingen bestaan, wat verschillend is van een fluxon diecorrespondeerd met een homokliene oplossing. Verder zijn de veranderingen in de gevondenoplossingscurves berekend voor het geval dat de ratio van de kritieke stromen van groter dan1, door 1, naar kleiner dan 1 gaat.


ix

Contents

Abstract v

Samenvatting vii

1 Introduction 11.1 History of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Fluxons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Analysis 72.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Travelling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Unperturbed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Perturbed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Solution in the First Junction . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Solution in the Second Junction . . . . . . . . . . . . . . . . . . . . . 16

3 Stability 213.1 The Spectral Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 The Essential Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 The Point Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Σp for ε = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 Σp near λ = 0 for ε 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.3 Σp near λ = ±i√

1− c2, ±i√

1−c2

J for ε 6= 0 . . . . . . . . . . . . . . . 30

4 Numerical Results 354.1 The Swihart Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Unequal Junctions (J > 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.1 Backbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.2 No backbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 (Almost) Identical Junctions (J = 1, J − 1 ¿ 1) . . . . . . . . . . . . . . . . 454.4 Unequal Junctions (J < 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Conclusion and Recommendations 515.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.4 Recommendations for Further Research . . . . . . . . . . . . . . . . . . . . . 53


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x Contents

A Travelling Wave Solutions of the Perturbed Sine-Gordon Equation 55A.1 Unperturbed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.2 Solution in the First Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.3 Solution in the Second Junction . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B The Spectral Problem 73B.1 The Essential Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73B.2 The Point Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79B.3 The Evans function near λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.4 The Evans function near λ = ±i

√1− c2, ±i

√1−c2

J . . . . . . . . . . . . . . . 87

C AUTO Program 93C.1 The AUTO Equations File (fname.c) . . . . . . . . . . . . . . . . . . . . . . . 93C.2 The AUTO Constants File (c.fname) . . . . . . . . . . . . . . . . . . . . . . . 97C.3 The AUTO Runs Executive File (fname) . . . . . . . . . . . . . . . . . . . . 99

List of Symbols 105

List of Figures 109

Bibliography 111

Internet Resources 112


1

Chapter 1

Introduction

This report treats the state of a fluxon in one of the two junctions of a two-fold stack of longJosephson junctions. To give a little bit insight on the behaviour of the physical phenomenain a Josephson junction, first a brief introduction of superconductivity is given. Next, theeffects predicted and discovered by Josephson and the related junctions are described.

1.1 History of Superconductivity

Superconductivity was first discovered in 1911 by the Dutch physicist Heike KamerlinghOnnes. In 1911 Onnes began to investigate the electrical properties of metals in extremelycold temperatures. It already had been known for many years that the resistance of metalsdecreased when they were cooled below room temperature, but it was not known what limitingvalue the resistance would approach if the temperature would be reduced to very close to 0K.

Figure 1.1 HeikeKamerlingh Onnes.

Some scientists, such as William Kelvin, believed that electronsflowing through a conductor would come to a complete halt as thetemperature approached absolute zero. Other scientists, includ-ing Onnes, suspected that a cold wire’s resistance would dissipate.This suspicion suggested that there would be a steady decrease inthe electrical resistance, allowing a better conduction of electricity.Therefore Onnes passed a current through a very pure mercury wireand measured its resistance as he steadily lowered the temperature.When he reached 4.2K the resistance, suddenly, became unmeasur-able. Current was flowing through the mercury wire without re-sistance and Onnes called this phenomenon of no resistance to anelectrical current originally supraconductivity. Nowadays, this phe-nomenon is referred to as superconductivity. For this discovery hewas awarded the Nobel Prize in 1913. In the following years, manymore materials, besides mercury, were found to be superconductiveif they were cooled to a temperature below a material dependentcritical threshold Tc.

However, it took over nearly half a century before an explanation of superconductivity wasfound. The first step in understanding superconductivity was made by Herbert Frohlichin 1950. He realized that, under the right conditions, electrons could have an attractiveinteraction mediated by phonons. These phonons are quanta of lattice vibration energy whichare analogous to photons, the quanta of light. Subsequently, in 1956 Leon Cooper showedthat under those right conditions, electrons could be bound in pairs. He showed that when


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2 1. Introduction

Figure 1.2 Various Josephson junctions.

a (negatively charged) electron moves through the lattice of the superconductor, the latticearound the electron distorts. This distortion causes positively charged phonons to be emitted,so there is an increase in the positive charge density around the electron. As a result, anotherelectron in the vicinity of this distortion is attracted to this greater positive charge densityand hence establishes a pair with the first electron. Such an electron pair is called a Cooperpair. So, the forces exerted by the phonons overcome the electron’s natural repulsion fromeach other. A year later, John Bardeen, Leon Cooper and John Schieffer developed theirTheories of Superconductivity, the BCS Theory. Because of the binding between them, theelectrons that form a Cooper pair are more resistent to vibrations in the lattice than a singleelectron would be, i.e. the attraction to the other electron will keep the pair “on course”.Therefore, Cooper pairs are able to move through the lattice, relatively unaffected by thethermal vibrations, below the critical temperature. So the current carried by the Cooperpairs experiences no (worth mentioning) resistance and is therefore called supercurrent andthe conducting medium a superconductor.

1.2 Josephson Junctions

In 1962, Brian Josephson predicted that, even though two superconductors were separatedby a thin non-superconducting barrier, it was possible to have a supercurrent from one super-conductor to another, the Josephson effect. Normally, when two metals are separated by avery small distance, quantum mechanics predicts that there is a small change that an electronbelonging to one of the metals can be found outside that metal. So, an electron can “jump”from the one metal to the other, which is called tunnelling. Now, if a potential difference isapplied to the metals, it is possible for a current to flow from one metal to the other. Joseph-son discovered that if two superconductors are placed very close to each other with in betweena thin non-superconducting layer, i.e. an insulator or a non-superconducting metal, Cooperpairs were able to tunnel through the barrier from one superconductor to another. Since this

Figure 1.3 Three stacks of long Josephson junctions.


1.2. Josephson Junctions 3

Figure 1.4 The Josephson penetration depth λJ of a Josephson junction.

tunnelling happened in the absence of a potential difference between the superconductors,he concluded that the tunnelling Cooper pairs could not experience any resistance. This isknown as the dc-Josephson effect and the configuration of the superconductors together withthe non-superconducting layer is called a Josephson junction. As long as the tunnelling su-percurrent is below the junction’s critical current (which depends on the material’s physicalproperties) there will be zero resistance and no voltage drop across the junction. However,as soon as the critical current is exceeded or a bias current is applied to the junction, also anormal current will flow across the junction. This is called the ac-Josephson effect, becausethe voltage related to this normal current across the junction depends on time, i.e. it is anac-voltage.

There are various types of Josephson junctions possible, as depicted in Figure 1.2. In Figure1.2(a) two superconductors are shown, where there is a non-superconducting layer in betweenthe endpoints of the superconductors. Another way of creating a Josephson junction is tocover a superconducting strip with a non-superconducting layer and put another strip acrosson top of the (covered) first one (Figure 1.2(b)). However, the Josephson junctions that areconsidered in this report are another type of junctions. They are (stacks of) long Josephsonjunctions. Figure 1.3 shows three different stacks of two long Josephson junctions. Thesejunctions are called long Josephson junctions because of the fact that the direction along thejunction is large compared to the Josephson penetration depth (λJ in Figure 1.4). When asuperconductor is placed in a magnetic field a current will start to flow along the surface ofthe conductor in order to induce a magnetic field opposite to the external field. Therefore,this external field penetrates the superconductor only over a short distance in which it decaysexponentially to zero, called the London penetration depth. So the induced current screens thebulk of the superconductor from the magnetic field. At the junctions, however, this screeningis imperfect, so the magnetic field can enter the junction over a certain distance. This distanceis the Josephson penetration depth. The third type of long Josephson junctions which isshown in Figure 1.3 are the annular junctions. Because of the absence of boundaries alongthe junctions, annular junctions are most often convenient for measurements and observations.


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4 1. Introduction

Figure 1.5 A fluxon in a two-fold stack of long Josephson junctions ([1|0] state).

1.3 Fluxons

In long Josephson junctions some special phenomena can occur. One of these phenomena,which is the most important for this report, is the existence of a fluxon. This fluxon is acirculating current across the non-superconducting layer of the Josephson junction, as shownin Figure 1.5. By applying a bias current (Ib in Figure 1.5) the fluxon can be forced to movealong the junction. For the study and analysis of fluxons, most often an uniform bias currentis used (because of simplicity, since in real physical system the bias current does not have tobe uniform). If the bias current is applied to the top of the junction, as shown in the figure,it will flow along the surface of the top superconductor and then vertically down throughthe junction and leave along the surface of the bottom superconductor. Since the circulatingcurrent induces a magnetic field, the fluxon will experience a Lorenz force by the verticallyflowing applied bias current. As a result, the fluxon starts to travel along the junction withincreasing speed until it reaches a (maximum) velocity for which there is a balance betweenthe dissipative effects of the junction and the strength of the (on the applied bias currentdepending) Lorenz force.The variety of fluxon configurations for a stack of Josephson junctions is very rich. For astack of N junctions, the notation [n1|n2|...|nN−1|nN ] stands for a configuration where thereare ni fluxons in the ith junction, called the [n1|n2|...|nN−1|nN ] state.

1.4 Applications

The possible application of Josephson junctions covers a wide area. This varies from theexploration of macroscopic quantum behaviour, as in macroscopic tunnelling experiments, tohigh-performance complex circuits involving thousands of elements, as in digital chips. Amain category of devices in which Josephson junctions are applied are so-called SQUIDs,Superconducting Quantum Interference Devices ([14]). These SQUIDs were invented in 1962after Brain Josephson’s findings. There are two (main) types of SQUIDs, the DC and RF (orAC) SQUIDs, where a RF SQUID consists of only one junction and a DC SQUID of two ormore junctions (see Figure 1.6). These SQUIDs are able to measure extremely tiny (changesof) magnetic fields and are probably the most sensitive devices existing for this purpose. Theadvantage of DC SQUIDs is that they are more sensitive than RF SQUIDs but they havea disadvantage that they are also more difficult and expensive to produce. Because of theirsensitivity, SQUIDs are well suited to study, e.g., neural activity inside brains but they arealso useful as precision movement sensors for, e.g., oil prospecting, earthquake prediction andgeothermal energy surveying. The operation of SQUIDs is based on the property of super-


1.4. Applications 5

Figure 1.6 A DC SQUID containing two Josephson junctions.

currents (besides their property of experiencing no resistance) that the magnetic flux passingthrough an area bounded by a supercurrent is quantized, i.e. the magnetic flux through anon-superconducting area induces a supercurrent in the superconducting boundary around itsuch that the total magnetic flux through the area becomes a multiple of the physical con-stant Φ0, the magnetic flux quantum. For this reason the circulating current in a Josephsonjunction is called fluxon, since it carries one quantum of magnetic flux.

Another important application of Josephson junctions is in the realization of digital circuits inwhich the two possible fluxon states in the junction represent the binary digits and in RSFQ(Rapid Single Flux Quantum) technology ([14], [16]). This RSFQ technology is based on thevery fast propagation of a fluxon along the junction. By relating (the passing at a certainposition of) the fluxon to a binary digit, an annular junction can operate at a very high clockfrequency which can be used as a clock to synchronize logical circuits operations.

The last application mentioned here is the use of Josephson junctions as a radiation source.If a moving fluxon reaches the edge of the junction radiation can be emitted on collisionwith the edge ([16]). This emitted radiation can be of very high frequencies but the powergenerated by a single junction is relatively low. One of the solutions to increase the outputpower is to connect many junctions in array configurations or use a stack of junctions whichare phase locked, i.e. the fluxons in the different junctions collide simultaneous with the edge.However, phase locking of the junctions can be rather difficult because of the sensitivity toexternal influences such as magnetic disturbance.


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7

Chapter 2

Analysis

In this chapter, first the modelling of a Josephson junction is discussed, starting from a singlejunction build up out of two superconductors and then step by step extended by incorporatingdissipative terms and the coupling between junctions. Subsequently, a perturbation analysisis performed, in order to determine the solutions of the unperturbed and perturbed systemin the first and second junction of a two-fold stack.

2.1 Modelling

As said, first the modelling of a stack of Josephson junctions is discussed. A complete deriva-tion of the equations can be found in [16, Chapter 2]. These equations are obtained by takingthe continuum limit of a discrete model for the junction configuration.

The modelling starts by considering two conductors in superconducting state (i.e. all electronshave formed Cooper pairs) with a non-superconducting layer in between. Since all electronsare in the same state (the ground state), they can be described by a single wave function foreach superconductor. These wave functions are complex valued functions and can be writtenas

ΨT =√

ρT eiΘT , ΨB =√

ρB eiΘB , (2.1)

where ΨT and ΨB are the wave functions for the top and the bottom superconductor, respec-tively. Furthermore, ρT (ρB) and ΘT (ΘB) are the density of the electron pairs and the phaseof the wave function in the top (bottom) conductor. The wave functions for the electrons inthe top and bottom superconductor satisfy the Schrodinger equations of the system, given bythe expressions

i~∂

∂tΨT = ET ΨT −KΨB,

i~∂

∂tΨB = EBΨB −KΨT ,

(2.2)

where K is a measure for the influence of one superconductor on the other. If a voltage isapplied to the junction, the zero level of the energy can be taken as the average of the potentialenergy in the top and the bottom superconductor, i.e. ET = −EL = −eV . Substituting thewave functions in the Schrodinger equations, by solving for the real and complex part thefollowing equations for the electron density in the top and bottom superconductor are obtained

∂ρT

∂t=

2~K√

ρT ρB sin(Φ)∂ρB

∂t= −2

~K√

ρT ρB sin(Φ), (2.3)


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8 2. Analysis

Figure 2.1 Flowing currents in the Josephson junction, arising from a (local) increase from0 to 2π in the phase difference between the superconductors.

where Φ is defined as Φ = ΘT − ΘB, the phase difference between the top and the bottomconductor. From these equations, it is observable that for Φ positive but less than π theelectron density in the top superconductor increases. So, there is a current flowing from thetop to the bottom superconductor, whereas for Φ larger than π and less than 2π there is acurrent flowing from the bottom to the top superconductor. Since a current also flows alongthe surface of the superconductors, an increase of the phase difference along the junction by2π gives rise to a circulating current (the fluxon), as depicted in Figure 2.1.

The Josephson junction can be modelled by the sine-Gordon equation. Using the capacitivelyshunted junction (CSJ) model, in which it is assumed that there is only a supercurrent acrossthe junction and an inductance for currents along the junction, the (unperturbed) sine-Gordonequation is obtained. This sine-Gordon equation is given by

∂2

∂x2Φ(x, t)− ∂2

∂t2Φ(x, t)− sin(Φ(x, t)) = 0. (2.4)

Subsequently the resistively and capacitively shunted junction (RCSJ) model is used to in-corporate non-zero resistance currents across the junction. These currents are due to quasi-particles (i.e. broken up Cooper pairs) tunnelling from one superconductor to the other. Thistunnelling arises if an (arbitrary small) potential difference between the top and the bottomsuperconductor is present. Now, the extended model for a single junction is given by

∂2

∂x2Φ(x, t)− ∂2

∂t2Φ(x, t)− sin(Φ(x, t)) = α

∂

∂tΦ(x, t)− γ. (2.5)

Here α describes the dissipative effect for the quasi-particle tunnelling, because of the insula-tor’s resistance experienced by the single electrons. The applied exterior bias current densityis reflected by γ, which is assumed to be uniform. The model can be extended even more byalso assuming a quasi-particle contribution in the superconductors. This (quasi-particle) flowalong the surface also experiences some resistance. Hence, the equation becomes

∂2

∂x2Φ(x, t)− ∂2

∂t2Φ(x, t)− sin(Φ(x, t)) = α

∂

∂tΦ(x, t)− β

∂

∂t

∂2

∂x2Φ(x, t)− γ, (2.6)

where β describes the dissipative effect because of the surface losses.

Next, considering a stack of two Josephson junctions, the complete configuration consistsof three superconductors which are separated by two non-superconducting layers, as shownin Figure 1.5. If the middle superconductor in the stack is thick, then the two Josephsonjunctions are decoupled. This means that they behave as two single Josephson junctionswhich do not influence each other. If on the contrary the middle superconductor is not sothick (i.e. smaller than the London penetration depth), then the two Josephson junctionswill influence each other. This happens because the magnetic field, induced by the currentsflowing at the top surface of the middle superconductor, can penetrate the superconductor


2.2. Travelling Waves 9

over this short distance and hence influence the currents flowing at the bottom surface of themiddle superconductor (and of course also the other way around). So, the phase differencesover the two junctions can influence each other if the separation between the junctions issmall. Taking this influence into account by using the RCSJ model, the equations for a stackof two junctions are given by (equation (2.3.2) in [16])

∂2

∂x2ΦA(x, t)− ∂2

∂t2ΦA(x, t)− sin(ΦA(x, t)) =

α∂

∂tΦA(x, t)− β

∂

∂t

∂2

∂x2ΦA(x, t) + S

∂2

∂x2ΦB(x, t)− γA

∂2

∂x2ΦB(x, t)−∆C

∂2

∂t2ΦB(x, t)−∆

sin(ΦB(x, t))J

=

∆Rα∂

∂tΦA(x, t)−∆Zβ

∂

∂t

∂2

∂x2ΦA(x, t) + ∆S

∂2

∂x2ΦB(x, t)− γB,

(2.7)

where ΦA and ΦB (γA and γB) are the phase differences (applied bias currents) in the topand bottom junction, respectively. S is the magnetic coupling parameter and the constants∆, C, J, R and Z are the ratios of the inductance, capacity, critical current, resistance andsurface resistance of the junctions, respectively (i.e. the ratios of the junctions physical mate-rial properties). If the two junctions are completely identical, these constants are (obviously)all equal to 1.

2.2 Travelling Waves

Although in general the ratio constants in equations (2.7) for the two junctions all can bedifferent, the focus in this report lies on systems in which only J might not be equal to 1, i.e.the junctions might only differ in their critical currents. Under this assumption, equations(2.7) simplify to

ΦAxx − ΦA

tt − sin(ΦA) = αΦAt − βΦA

xxt + SΦBxx − γA

ΦBxx − ΦB

tt −sin(ΦB)

J= αΦB

t − βΦBxxt + SΦA

xx − γB.(2.8)

In this model, the spatial coordinate x is measured in units of the Josephson penetrationdepth λJ , the time t in units of the inverse Josephson plasma frequency ω−1

0 , γA (γB) in unitsof the junctions critical current density jA

c (jBc ) and J is the ratio of the two critical current

densities (J = jAc /jB

c ). By measuring x and t in these units, the constants α and β whichdescribe the damping and the constant S which describes the magnetic coupling, come outdimensionless. Furthermore are the damping constants assumed to be positive, the magneticcoupling S negative between −1 and 0 and 0 ≤ γA, γB ≤ 1 (actually |γA|, |γB| ≤ 1, wherethe sign of γA and γB is determined by the direction in which the bias current is applied;positive for a current from the top to the bottom superconductor and negative for a currentin the other direction).

From here on, the applied bias current densities of the first and the second junction areassumed to be equal. The phase difference over the first junction is represented by ΦA,where the phase difference over the second junction is represented by ΦB. Furthermore, thequasi-particle flow along the surface of the superconductors is neglected, i.e. β = 0. Thedissipative effect for the quasi-particles tunnelling through the non-superconducting layer issmall in practice. Therefore it is scaled by ε, as well as the other constants S and γ, i.e.α = εα, S = εS, γ = εγ. Although S and γ do not need to be small, they are taking small


June 2005

10 2. Analysis

to be able to perform a perturbation analysis. In Section 1.3 it was mentioned that a fluxoncould travel along the junction. Therefore it is reasonable to look for travelling wave solutionsto the equations (2.8). By defining the travelling wave coordinate ξ as

ξ =x− ct√1− c2

, |c| < 1, (2.9)

the solution is assumed to depend only on ξ, i.e. ΦA,B(x, t) = ϕA,B( x−ct√1−c2

) = ϕA,B(ξ). Theequations (2.8) now reduce to ordinary differential equations for the travelling wave solutionswhich are given by

ϕAξξ − sin(ϕA) = ε

(− αc√

1− c2ϕA

ξ +S

1− c2ϕB

ξξ − γ

)

ϕBξξ −

sin(ϕB)J

= ε

(− αc√

1− c2ϕB

ξ +S

1− c2ϕA

ξξ − γ

),

(2.10)

with the tildes on α, S and γ dropped.

2.3 Unperturbed System

In this and the following sections the travelling wave solution in the first and the secondjunction is determined. First the solutions of the unperturbed system is determined and sub-sequently the solutions of the system under a small perturbation, i.e. ε 6= 0. A more detailedderivation of the solutions can be found in Appendix A.

Considering the system without the small perturbation terms (i.e. ε = 0). The resultingsystem consists of two uncoupled homogeneous sine-Gordon equations for the phase differencesin the first and the second junction

ϕAξξ − sin(ϕA) = 0,

ϕBξξ −

sin(ϕB)J

= 0.(2.11)

A particularly important solution of the homogeneous sine-Gordon equation ϕξξ−sin(ϕ) = 0

Figure 2.2 Phaseportrait of the homogeneous sine-Gordon equation in the (ϕ,ϕξ)-plane.


2.4. Perturbed System 11

Figure 2.3 The soliton solution of the unperturbed sine-Gordon equation, which correspondsto the fluxon in a Josephson junction.

which corresponds to an increase of the phase by 2π, i.e. under the conditions limξ→−∞ ϕ(ξ) =0 and limξ→∞ ϕ(ξ) = 2π, is the 2π-kink (also called soliton) solution given by

ϕ0(ξ) = 4 arctan( eξ). (2.12)

This soliton solution corresponds to the orbit connecting the fixed points (ϕ, ϕξ) = (0, 0) and(ϕ,ϕξ) = (2π, 0), which is denoted in Figure 2.2 by the thick curve. Since the variable ϕ is aphase variable, an increase of ϕ by 2π has no effect on the system. Hence, the solution con-necting the equilibria (0, 0) and (2π, 0) in the phaseportrait can be viewed of as a homoclinicorbit. In a Josephson junction without quasi-particle tunnelling and no applied bias currentthis soliton solution represents the fluxon.

For the situation of a stack of two junctions in the [1|0] state, i.e. the state where there isone fluxon in the first junction and no fluxon in the second, this implies that the solutionsof the uncoupled homogeneous equations (2.11) are given by ϕA = ϕA

0 (ξ) = 4 arctan( eξ) andϕB = ϕB

0 (ξ) = 0. In Figure 2.3 the soliton solution of the first junction is shown, from whichthe 2π-jump is clearly observable.

2.4 Perturbed System

Next, a small perturbation in the system is considered, i.e. 0 < ε ¿ 1 in equations (2.10).The fixed points of the perturbed system can be divided in the classes

ϕA = ϕA2n = arcsin(εγ) + 2nπ,

ϕB = ϕB2n = arcsin(εγJ) + 2nπ,

n = 0,±1,±2, ... (2.13)

andϕA = ϕA

2n+1 = π − arcsin(εγ) + 2nπ,ϕB = ϕB

2n+1 = π − arcsin(εγJ) + 2nπ,n = 0,±1,±2, ...., (2.14)

where ϕA,Bξ = 0. Looking for travelling wave solutions of the perturbed system which corre-

spond to the [1|0] state now implies finding solutions ϕA(ξ), which connects the fixed points(ϕA

2n, 0), (ϕA2(n+1), 0) and ϕB(ξ), which connects the fixed points (ϕB

2n, 0), (ϕB2n, 0) (since there

is no fluxon in the second junction, there is also no increase of the phase by 2π and henceϕB should connect the same fixed points). Solutions to the perturbed system (2.10) can befound by looking for solutions in the form of series in ε,

ϕA,B(ξ) = ϕA,Bε (ξ) = ϕA,B

0 (ξ) + εϕA,B1 (ξ) +O(ε2), (2.15)


June 2005

12 2. Analysis

where the ϕA,B1 (ξ) are bounded on R. Furthermore ϕA

ε (ξ) should connect the fixed points(ϕA

0 , 0), (ϕA2 , 0) and ϕA

ε (0) is imposed to be π + arcsin(εγ) to get rid of an arbitrary phaseshift whereas ϕB

ε (ξ) should connect the fixed points (ϕB0 , 0), (ϕB

0 , 0). Now ϕA,B0 (ξ) are given

by the solutions of the (uncoupled) unperturbed system, as in Section 2.3. By substitution ofthe series (2.15) in the perturbed equations and making the transformations ϕA

1 = ϕA1 +γ and

ϕB1 = ϕB

1 + γJ , the systems for ϕA,B1 (ξ) are obtained by collecting terms of order ε. Hence,

for ϕA1 the equation

(ϕA1 )ξξ − cos(ϕA

0 )ϕA1 = − αc√

1− c2(ϕA

0 )ξ − γ(1− cos(ϕA0 )) (2.16)

must be solved under the conditions

limξ→±∞

ϕA1 (ξ) = 0 with ϕA

1 (0) = 0 (2.17)

and for ϕB1 (ξ) the equation

(ϕB1 )ξξ − ϕB

1

J=

S

1− c2(ϕA

0 )ξξ (2.18)

must be solved under the conditions

limξ→±∞

ϕB1 (ξ) = 0. (2.19)

These boundary conditions for ϕA,B1 (ξ) follow from the values of the fixed points that ϕA,B

ε (ξ)should connect, i.e. limξ→−∞ ϕA

ε (ξ) = ϕA0 , limξ→∞ ϕA

ε (ξ) = ϕA2 and limξ→±∞ ϕB

ε (ξ) = ϕB0 .

The equations (2.16) and (2.18) can be rewritten in (first order differential) matrix equations,resulting in

(ΨA1 )ξ = AA(ξ)ΨA

1 + HA(ξ) and

(ΨB1 )ξ = ABΨB

1 + HB(ξ),(2.20)

where ΨA,B1 = [ϕA,B

1 (ξ), (ϕA,B1 (ξ))ξ]T (note that AB is ξ-independent).



2.4.1 Solution in the First Junction

To obtain the travelling wave solution of order ε in the first junction, the equation for ΨA(ξ)in (2.20) must be solved for ϕA

1 (ξ) under the conditions (2.17). The matrices AA(ξ) andHA(ξ) in the equation are given by

AA(ξ) =[

0 1cos(ϕA

0 (ξ)) 0

]and HA(ξ) =

0

− αc√1−c2

(ϕA0 (ξ))ξ +

− γ(1− cos(ϕA0 (ξ)))

. (2.21)

where ϕA0 (ξ) is the solution of the unperturbed system in the first junction, i.e. ϕA

0 (ξ) =4 arctan( eξ).

First, the homogeneous equation (ΨA1 )ξ = AA(ξ)ΨA

1 is considered. Let X(ξ) be a fundamentalmatrix solution of the homogeneous equation, i.e. X(ξ) satisfies Xξ(ξ) = AA(ξ)X(ξ). Thecolumns of X(ξ) consist of linearly independent solutions of the homogeneous equation. Sincethis homogeneous system for ϕA

1 (ξ) equals the linearization of equation (2.11)1 around ϕA0 (ξ),

the homogeneous system for ϕA1 (ξ) is solved by the derivative of ϕA

0 (ξ) (up to a constant).Hence, the first column of X(ξ) is given by [(ϕA

0 )ξ, (ϕA0 )ξξ]T (divided by 2). The second

column of the fundamental matrix follows from the fact that the solutions should be linearlyindependent, i.e. det(X(ξ)) 6= 0. Since the derivative of determinant with respect to ξ iszero this implies that the determinant of X(ξ) is a constant. This constant can by takingequal to 1, in accordance with Liouville’s theorem of phase conservation. Hence, the secondcolumn can be constructed in such a way that for ξ = 0 it holds that X(0) = I2 (i.e. X(ξ)is the principal matrix solution of the homogeneous equation at initial “time” ξ = 0). Afterdetermining the second column, the principal matrix solution of the homogeneous equationis given by

X(ξ) =[

sech(ξ) 12(ξ sech(ξ) + sinh(ξ))

( sech(ξ))′ 12(ξ sech(ξ) + sinh(ξ))′

], (2.22)

where the prime denotes differentiating with respect to ξ. Looking at the asymptotic be-haviour of the elements of X(ξ) for ξ → ±∞, it holds that

limξ→±∞

X1,1 = 0, limξ→±∞

X1,2 = ±∞,

limξ→±∞

X2,1 = 0, limξ→±∞

X2,2 = ∞.(2.23)

This shows that the elements in the first column are bounded for ξ → ±∞.

In order to find a bounded solution for (2.20)1, projections on the stable and unstable subspaceare defined. The stable subspace is given by Ssub = {p ∈ R2 | supt≥0 |X(t)p| < ∞}, i.e.those (initial) vectors p such that the linear combination of the independent solutions inX(ξ) is bounded for ξ ∈ R+. Likewise, the unstable subspace is given by Usub = {p ∈R2 | supt≤0 |X(t)p| < ∞}, i.e. the vectors p such that the linear combination of the solutionsis bounded for ξ ∈ R−. Using the asymptotic behaviour of the element of X(ξ), the projectionsP s(0) on the stable and P u(0) on the unstable subspace, at ξ = 0, can be defined as follows

P s(0) =[

1 00 0

]= P u(0). (2.24)

Furthermore, P u(0) = I2 − P s(0) and P s(0) = I2 − P u(0) are defined, so P u(0) = P s(0).Hence, with the projection P s(0), the equation (ΨA

1 )ξ = AA(ξ)ΦA1 possesses an exponential


June 2005

14 2. Analysis

dichotomy on R+. Similarly, the equation possesses an exponential dichotomy on R− with theprojection P u(0). Having an exponential dichotomy on R+ means that the principal matrixsolution X(ξ) of the homogeneous equation and the projection P s(0) on the stable subspacesatisfy

{ |X(t)P s(0)X−1(s)| ≤ K e−α(t−s), t ≥ s ≥ 0, K, α > 0,

|X(t)(I2 − P s(0))X−1(s)| ≤ L e−β(s−t), s ≥ t ≥ 0, L, β > 0.(2.25)

For the exponential dichotomy on R− similar relations are satisfied with the principal matrixsolution and the projection P u(0) on the unstable subspace. Because the homogeneous equa-tion possesses an exponential dichotomy on R+ and R− and limξ→±∞HA(ξ) = [0, 0]T , theinhomogeneous equation for ΨA

1 (ξ) of (2.20) has a bounded solution on R+ and on R− ([6]).If both these bounded solutions also match at ξ = 0, then there exists a bounded solution ofthe inhomogeneous equation on the whole R.

Bounded Solution for ξ ≥ 0

The general solution of the inhomogeneous equation (2.20)1 is found by applying the variation-of-constants formula, i.e.

ΨA1 (ξ) = X(ξ)

[X−1(σ)ΨA

1 (σ) +∫ ξ

σX−1(τ)HA(τ) dτ

]. (2.26)

In order to (first) determine a bounded solution ΨA1

+(ξ) on R+, the projections P s(ξ) =X(ξ)P s(0)X−1(ξ) and P u(ξ) = X(ξ)P u(0)X−1(ξ) are defined, i.e. P s(ξ) is the projection onthe stable subspace for ξ ≥ 0. Since these projections satisfy P s(ξ)+P u(ξ) = I2, the solutionΨA

1+(ξ) can be split up as follows

ΨA1

+(ξ) = P s(ξ)ΨA

1+(ξ) + P u(ξ)ΨA

1+(ξ)

= X(ξ)P s(0)X−1(ξ)ΨA1

+(ξ) + X(ξ)P u(0)X−1(ξ)ΨA

1+(ξ), ξ ≥ 0,

(2.27)

where ΨA1

+(ξ) is of the form (2.26). Since P s(0) is the projection on the stable manifold(at ξ = 0), the solution ΨA

1+(ξ) is bounded on R+ if the unstable contribution is zero.

Furthermore, since σ is free to choose in equation (2.26), by substituting the general solutionΨA

1+(ξ) = X(ξ)[X−1(σ)ΨA

1+(σ)+

∫ ξσ X−1(τ)HA(τ) dτ ] in the last expression’s right hand side

and taking σ = 0 it follows that the unstable contribution to the solution is zero (and henceΨA

1+(ξ) is bounded on R+) if

P u(0)ΨA1

+(0) = −

∫ ∞

0P u(0)X−1(τ)HA(τ) dτ (2.28)

is satisfied (here is made use of the fact that X−1(0) = I2). This results in the condition

(ϕA1

+)ξ(0) =

4αc + γπ√

1− c2

2√

1− c2, (2.29)

by working out the integrals. Provided that the condition (2.28) is satisfied, the bounded so-lution on R+ is explicitly given by writing down the expression for the two terms P s(ξ)ΨA

1+(ξ)

and P u(ξ)ΨA1

+(ξ) of (2.27). This means that the first term turns into

P s(ξ)ΨA1

+(ξ) = X(ξ)P s(0)ΨA

1+(0) + X(ξ)

∫ ξ

0P s(0)X−1(τ)HA(τ) dτ, ξ ≥ 0 (2.30)



by choosing σ = 0. Similarly by choosing σ = ∞, the second term of (2.27) is given by

P u(ξ)ΨA1

+(ξ) = X(ξ)

∫ ξ

∞P u(0)X−1(τ)HA(τ) dτ, ξ ≥ 0 (2.31)

since X(ξ)P u(0)X−1(σ)ΨA1

+(σ) will become zero if σ goes to ∞ for a bounded solution onR+. This is because of the exponential dichotomy of the system on R+. Hence, the boundedsolution ΨA

1+(ξ) is the sum (of the right-hand sides) of (2.30) and (2.31).

Bounded Solution for ξ ≤ 0

Likewise for ξ < 0, in order to determine a bounded solution ΨA1−(ξ) on R−, the projections

P u(ξ) = X(ξ)P u(0)X−1(ξ) and P s(ξ) = X(ξ)P s(0)X−1(ξ) are defined. By following thesame procedure as on R+, the bounded solution on R− is explicitly given by

ΨA1−(ξ) = X(ξ)P u(0)ΨA

1−(0) + X(ξ)

∫ ξ

0P u(0)X−1(τ)HA(τ) dτ +

+ X(ξ)∫ ξ

−∞P s(0)X−1(τ)HA(τ) dτ, ξ ≤ 0,

(2.32)

provided that the condition

P s(0)ΨA1−(0) = −

∫ −∞

0P s(0)X−1(τ)HA(τ) dτ (2.33)

is satisfied, which results in the condition

(ϕA1−)ξ(0) =

−4αc− γπ√

1− c2

2√

1− c2. (2.34)

Bounded Solution for ξ ∈ RA bounded solution of (ΨA

1 )ξ = AA(ξ)ΨA1 + HA(ξ) on the whole R now exists if the bounded

solution ΨA1

+(ξ) on R+ and the bounded solution ΨA1−(ξ) on R− match at ξ = 0. This means

that ΨA1

+(0) = ΨA1−(0) should hold, which is equivalent to

P s(0)ΨA1

+(0) +

∫ 0

∞P u(0)X−1(τ)HA(τ) dτ

= P u(0)ΨA1−(0) +

∫ 0

−∞P s(0)X−1(τ)HA(τ) dτ,

(2.35)

by evaluating the expression of ΨA1

+(ξ) and ΨA1−(ξ) at ξ = 0 and setting them equal to each

other. Since P s(0) = P u(0) and P s(0) = P u(0) the matching requirement turns into therequirements P s(0)ΨA

1+(0) = P s(0)ΨA

1−(0) and

∫ ∞

−∞P u(0)X−1(τ)HA(τ) dτ = 0. (2.36)

The former requirement gives ϕA1

+(0) = ϕA1−(0) (= ϕA

1 (0)) where ϕA1 (0) is taken 0, in accor-

dance with the second mentioned condition in (2.17), to get rid of the translation invariance.The latter requirement results in −4αc+γπ

√1−c2√

1−c2= 0 and thus

c(γ) =−γπ√

(4α)2 + (γπ)2. (2.37)


June 2005

16 2. Analysis

Hence, for small perturbations the relation between the velocity c of the fluxon and the ap-plied bias current density γ of a bounded solution on R is given by (2.37). Furthermore,because of this relation it follows that (ϕA

1+)ξ(0) = 0 = (ϕA

1−)ξ(0) (= (ϕA

1 )ξ(0)).

By using relation (2.37) and ΨA1 (0) = [0, 0]T , the bounded solution of (2.20)1 on R can be

obtained. In integral form the solution is of the form

ΨA1 (ξ) =

X(ξ)[∫ ξ

−∞P s(0)X−1(τ)HA(τ) dτ −

∫ 0

ξP u(0)X−1(τ)HA(τ) dτ

], ξ < 0,

X(ξ)[∫ ξ

0P s(0)X−1(τ)HA(τ) dτ −

∫ ∞


], ξ ≥ 0,

(2.38)

where HA(τ) = HA(τ)|c=c(γ), i.e. relation (2.37) substituted in HA(τ). The integrals can beworked out, resulting in the explicit form for ϕA

1 (ξ),

ϕA1 (ξ) = γ sech(ξ)

[( eξ − 1)2

e2ξ + 1− πξ( e2ξ − 1)

4( e2ξ + 1)+

∫ ξ

0

8τ e3τ

( e2τ + 1)3

]+

+12γ(ξ sech(ξ) + sinh(ξ))

[π e2ξ

e2ξ + 1+

2( eξ − e3ξ)( e2ξ + 1)2

− 2 arctan( e2ξ)]

, ξ ∈ R,

(2.39)

which is the expression for ξ ∈ R, since the expressions of ΨA1 (ξ) are the same for ξ < 0 and

ξ ≥ 0. In fact, this is not surprising since the stable direction (for ξ ≥ 0) and the unstabledirection (for ξ ≤ 0) are the same. Moreover, since there is no S involved in the expressionof ϕA

1 (ξ), this implicates that the coupling between the junctions does not influence the O(ε)term of ϕA

ε (ξ). This is due to the fact that ϕB0 (ξ) = 0, since if it was not equal to 0 it would

give a term of O(ε) in the perturbed differential equation for ϕA(ξ). Hence, the effect of thesecond junction on the first junction will be visible in the O(ε2) (and higher) terms of ϕA

ε (ξ).

2.4.2 Solution in the Second Junction

As in the first junction, for the travelling wave solution in the second junction of order ε,equation (2.20)2 must be solved under the conditions (2.19). This means finding a solutionΨB

1 (ξ) to the equation (ΨB1 )ξ = ABΨB

1 +HB(ξ), where the matrices AB and HB(ξ) are givenby

AB =

[0 11J

0

]and HB(ξ) =

0S

1− c2(ϕA

0 (ξ))ξξ

. (2.40)

Since the matrix AB is a constant matrix, the fundamental matrix solution Y (ξ) of thehomogeneous equation (ΨB

1 )ξ = ABΨB1 is the exponential matrix of AB, i.e. Y (ξ) = eABξ.

Moreover, Y (ξ) is the principal matrix solution of the homogeneous equation at initial “time”ξ = 0 (since Y (0) = I2) and is given by

Y (ξ) =

12

e−ξ√J +

12

eξ√J −

√J

2e−

ξ√J +

√J

2e

ξ√J

− 12√

Je−

ξ√J +

12√

Je

ξ√J

12

e−ξ√J +

12

eξ√J

(2.41)

The stable and unstable direction of this system are [1,−1/√

J ]T and [1, 1/√

J ]T for ξ ∈ R,so, as for the first junction, the projections P s(0) on the stable and P u(0) on the unstable



Figure 2.4 ϕA1 (ξ) (solid line) and (ϕA

1 )ξ(ξ) (dotted line) for γ = 1.

directions at ξ = 0 can be defined. With P u(0) = I2 − P s(0) and P s(0) = I2 − P u(0) theprojections are

P s(0) =

12

−√

J

2− 1

2√

J

12

and P u(0) =

12

√J

21

2√

J

12

= I2 − P s(0), (2.42)

so P s(0) = P s(0) and P u(0) = P u(0). Now, the equation (ΨB1 )ξ = ABΨB

1 possesses anexponential dichotomy on the whole real line with the projection P s(0), i.e.

{ |Y (t)P s(0)Y −1(s)| ≤ K e−α(t−s), t ≥ s, t, s ∈ R, K, α > 0,

|Y (t)P u(0)Y −1(s)| ≤ L e−β(s−t), s ≥ t, s, t ∈ R, L, β > 0.(2.43)

Analogue to the previous section, since limξ→±∞HB(ξ) = [0, 0]T , a bounded solution ofthe inhomogeneous equation (ΨB

1 )ξ = ABΨB1 + HB(ξ) on R will be constructed by finding

the bounded solutions on R+ and R− and match them at ξ = 0. For this, the projectionsP s(ξ) = Y (ξ)P s(0)Y −1(ξ) and P u(ξ) = Y (ξ)P u(0)Y −1(ξ) are defined. Then the solutionson R+ and R− can be split up like ΨB

1±(ξ) = P s(ξ)ΨB

1±(ξ) + P u(ξ)ΨB

1±(ξ), resulting in the

conditions that ΨB1

+(ξ) is bounded on R+ and ΨB1−(ξ) on R− provided that

P u(0)ΨB1

+(0) = −

∫ ∞

0P u(0)Y −1(τ)HB(τ) dτ

and

P s(0)ΨB1−(0) = −

∫ −∞

0P s(0)Y −1(τ)HB(τ) dτ

(2.44)


June 2005

18 2. Analysis

is satisfied, respectively. Evaluation of these expression gives

12ϕB

1+(0) +

√J

2(ϕB

1+)ξ(0) =

∫ ∞

0−1

2

√J e−

τ√J

S

1− c2(ϕA

0 (τ))ττ dτ

12ϕB

1−(0)−

√J

2(ϕB

1−)ξ(0) =

∫ 0

−∞−1

2

√J e

τ√J

S

1− c2(ϕA

0 (τ))ττ dτ

(2.45)

Provided the conditions (2.44) are satisfied, i.e. ΨB1

+ is bounded on R+ and ΨB1− is bounded

on R−, the split up terms of the bounded solutions can be simplified by using the fact that thesystem possesses an exponential dichotomy on R (and thus on R+ and R−). Subsequently,matching the resulting solutions at ξ = 0, i.e. solving ΨB

1+(0) = ΨB

1−(0) (= ΨB

1 (0)) resultsin the requirements

ϕB1 (0) = 0 and (ϕB

1 )ξ(0) = −∫ ∞

0e−

τ√J

S

1− c2(ϕA

0 (τ))ττ dτ, (2.46)

which satisfy (2.44). So, with this given ϕB1 (0) and (ϕB

1 )ξ(0), the solutions ΨB1

+(ξ) andΨB

1−(ξ) are bounded on R+ and R−, respectively. Furthermore, they are the same at ξ = 0,

i.e. the bounded solution of (ΨB1 )ξ = ABΨB

1 + HB(ξ) on R is of the form ΨB1−(ξ) for ξ < 0

and ΨB1

+(ξ) for ξ ≥ 0,

ΨB1 (ξ) =

Y (ξ)[P u(0)ΨB

1 (0) +∫ ξ

−∞P s(0)Y −1(τ)HB(τ) dτ +

−∫ 0

ξP u(0)Y −1(τ)HB(τ) dτ

], ξ < 0,

Y (ξ)[P s(0)ΨB

1 (0) +∫ ξ

0P s(0)Y −1(τ)HB(τ) dτ +

−∫ ∞


], ξ ≥ 0,

(2.47)

where HB(τ) is HB(τ) with relation (2.37) substituted for c. By evaluating this matrix form,the explicit form of ϕB

1 (ξ) can be obtained and is given by

ϕB1 (ξ) =

√J e

ξ√J

∫ 0

ξsinh(

τ√J

)S

1− c(γ)2(ϕA

0 (τ))ττ dτ +

+√

J sinh(ξ√J

)∫ ξ

−∞e

τ√J

S

1− c(γ)2(ϕA

0 (τ))ττ dτ, ξ < 0,

−√

J e−ξ√J

∫ ξ

0sinh(

τ√J

)S

1− c(γ)2(ϕA

0 (τ))ττ dτ +

−√

J sinh(ξ√J

)∫ ∞

ξe−

τ√J

S

1− c(γ)2(ϕA

0 (τ))ττ dτ, ξ ≥ 0.

(2.48)



Figure 2.5 ϕB1 (ξ) (solid line) and (ϕB

1 )ξ(ξ) (dotted line) for γ = 1, α = 1, S = −2 andJ = 1.


June 2005

21

Chapter 3

Stability

In this chapter the stability of the travelling wave solution in the first and the second junctionis considered. The stability of the solutions is investigated by looking at the essential spectrumand the point spectrum of the spectral problem. A more detailed derivation of the analysiscan be found in Appendix B.

3.1 The Spectral Problem

Consider the partial differential equations (2.8) (with β = 0). Since these equations ad-mit travelling wave solutions, corresponding to the [1|0] state, the equations are writtenin a moving coordinate frame. This is done by making the transformation ξ = x−ct√

1−c, i.e.

ΦA,B(x, t) = ΦA,B(ξ, t). In doing so, the equations become partial differential equations in tand in the travelling wave coordinate ξ,

ΦAξξ + 2c√

1−c2ΦA

ξt − ΦAtt − sin(ΦA) = ε

(− αc√

1−c2ΦA

ξ + αΦAt + S

1−c2ΦB

ξξ − γ)

,

ΦBξξ + 2c√

1−c2ΦB

ξt − ΦBtt − 1

J sin(ΦB) = ε(− αc√

1−c2ΦB

ξ + αΦBt + S

1−c2ΦA

ξξ − γ)

.(3.1)

This system is linearized around the travelling wave solutions ϕAε (ξ) and ϕB

ε (ξ), which wereintroduced in Section 2.4 (equation 2.15). This means that ΦA,B(ξ, t) are taken of the formϕA,B

ε (ξ) + uA,B(ξ, t), where uA,B(ξ, t) are assumed to be (initially) small. By determiningthe behaviour of uA,B(ξ, t) over time, it can be seen how solutions to the partial differentialequations (2.8) behave that are initially close to the travelling wave solutions, i.e. whathappens if a small perturbation arises in the travelling wave solutions. Now, ΦA,B(ξ, t) =ϕA,B

ε (ξ)+uA,B(ξ, t) is substituted in the equations (3.1) and the linearized partial differentialequations for uA,B(ξ, t) are obtained. The first equation of (3.1) turns into

uAξξ +

2c√1− c2

uAξt − uA

tt − cos(ϕAε )uA = ε

(− αc√

1− c2uA

ξ + αuAt +

S

1− c2uB

ξξ

), (3.2)

where the second equation is of similar form as this one, with A replaced by B (and vice versa)and the cosine term divided by J . Subsequently, the spectral Ansatz uA,B(ξ, t) = eλtvA,B(ξ)is made. Hence, the resulting system consist of ordinary differential equations of vA,B(ξ) withλ as parameter. This system can be rewritten as a first order differential matrix equation ofthe form

∂

∂ξV (ξ) = Aε(ξ, λ)V (ξ), (3.3)


June 2005

22 3. Stability

where V (ξ) = [vA(ξ), vAξ (ξ), vB(ξ), vB

ξ (ξ)]T and

Aε(ξ, λ) =

0 1 0 0ω(ξ,λ)1−ν2

η(λ)1−ν2

νχ(ξ,λ)1−ν2

νη(λ)1−ν2

0 0 0 1νω(ξ,λ)1−ν2

νη(λ)1−ν2

χ(ξ,λ)1−ν2

η(λ)1−ν2

, (3.4)

with

ω(ξ, λ) = λ2 + εαλ + cos(ϕAε (ξ)),

χ(ξ, λ) = λ2 + εαλ + 1J cos(ϕB

ε (ξ)),η(λ) = −2cλ−εαc√

1−c2and ν = ε S

1−c2

(3.5)

In this way, the stability of the travelling wave solutions around which is linearized is de-termined by the spectrum of the operator belonging (3.3). The operator belonging to theobtained eigenvalue problem is defined by T (λ) : U → (d/dξ − Aε(ξ, λ))U . The spectrum ofthis operator is given by those λ for which T (λ) is not invertible ([12]), i.e. those λ for whichthere exists a (non trivial) solution of (3.3). The complete spectrum of T (λ) is the union ofthe essential spectrum (also called continuous spectrum) and the point spectrum, where theessential and the point spectrum are disjunct.

The essential spectrum Σess consists of those λ for which the solutions of the eigenvalueproblem (3.3) are oscillating solutions. Defining (Aε)±∞(λ) = limξ→±∞Aε(ξ, λ), the bound-ary of the essential spectrum is then given by those λ for which the solutions of Vξ(ξ) =(Aε)±∞(λ)V (ξ) are oscillating solutions, i.e. vA,B(ξ) are of the form CA,B eiκξ with CA,B

multiplicative constants and κ real. In this way, the boundary of the essential spectrum isrelated to the stability of the equilibria as a result of using (Aε)±∞(λ). Analyzing (3.3) withAε(ξ, λ) = (Aε)±∞(λ) is in fact analyzing the linearization around the equilibrium points(the state to which the travelling wave solutions converge for ξ → ±∞). Since ϕA

ε (ξ) is a2π-kink solution and ϕB

ε (ξ) corresponds to the “0 state”, it holds that limξ→∞ cos(ϕA,Bε (ξ)) =

limξ→−∞ cos(ϕA,Bε (ξ)). As a consequence of this, for (Aε)±∞(λ) it holds that (Aε)+∞(λ) =

(Aε)−∞(λ) ≡ (Aε)∞. Now, if a part of the boundary of the essential spectrum (and hencea part of the essential spectrum) lies in the right half-plane of the complex plane, i.e. thereexist λ ∈ Σess with <(λ) > 0, this gives rise to an instability of the equilibrium points,since (an initially small) perturbation uA,B(ξ, t) = eλtvA,B(ξ) to the travelling wave solutionswill grow exponentially over time (this instability often manifests itself by the appearance ofsmall spatially periodic ripples on the asymptotic state of the travelling wave solutions ([16])).

The point spectrum Σp consists of those λ for which there exists a solution to (3.3) (i.e.an eigenfunction), such that also the boundary conditions limξ→±∞ V (ξ) = 0 are satisfied.Again, if the λ for which this solution exists has a real part greater than zero, this gives riseto an instability. Note that for λ = 0 the spectral Ansatz turns into uA,B(ξ, t) = vA,B(ξ).Hence, the linearized equations are simply solved by the derivatives of the travelling wavesolutions, i.e. vA,B(ξ) = (ϕA,B

ε )ξ(ξ). Since these derivatives satisfy the boundary conditions,i.e. limξ→±∞(ϕA,B

ε )ξ(ξ) = 0, the point spectrum contains at least the value λ = 0, where thecorresponding eigenfunction is given by V (ξ) = [(ϕA

ε )ξ, (ϕAε )ξξ, (ϕB

ε )ξ, (ϕBε )ξξ]T .


3.2. The Essential Spectrum 23

3.2 The Essential Spectrum

As mentioned, the boundary Σess of the essential spectrum is given by those λ in the spectrumof T (λ) for which there exists a solution of

∂ξ

vA(ξ)(vA)ξ(ξ)vB(ξ)

(vB)ξ(ξ)

= (Aε)∞(λ)

vA(ξ)(vA)ξ(ξ)vB(ξ)

(vB)ξ(ξ)

, (3.6)

where vA,B(ξ) are of the form vA,B(ξ) = CA,B eiκξ with κ real, i.e. oscillating solutions of(3.3). If for simplification the transformation λ = µ + iκc√

1−c2is used, it follows that

Σess ={

λ = µ + icκ√1−c2

∣∣∣(µ2 + εαµ + κ2

1−c2+

√1− ε2γ2

)×

×(

µ2 + εαµ +κ2

1− c2+

1J

√1− ε2γ2J2

)−

(εκ2S

1− c2

)2

= 0, κ ∈ R}

(3.7)

and the relation between the multiplicative coefficients CA and CB of the oscillating solutions(due to the coupling between the junctions) is given by

CB = −νω∞(λ) + iκη(λ)

κ2(1− ν2) + χ∞(λ) + iκη(λ)CA. (3.8)

Here ω∞(λ) and χ∞(λ) are the terms in (Aε)∞(λ) corresponding to ω(ξ, λ) and χ(ξ, λ) inAε(ξ, λ). Now, the boundary of the essential spectrum is given by those λ = µ+ iκc√

1−c2, where

µ is a root of an equation of the form (µ2 + a1µ + b1)(µ2 + a1µ + b2)− c21. Such a polynomial

of degree 4 is solved by µ = −12a1 ±(1)

12

√a2

1 ±(2) 2√

(b1 − b2)2 + 4c21 − 2(b1 + b2), where all

four possible combinations of ±(1) and ±(2) give a root, i.e.

µ(κ) = −12εα±(1)

12

√w±(2) , (3.9)

where

w±(2) = ε2α2 ±(2) 2

√(√1− ε2γ2 − 1

J

√1− ε2γ2J2

)2

+ 4(

εκ2S

1− c2

)2

− 2(

2κ2

1− c2+

√1− ε2γ2 +

1J

√1− ε2γ2J2

).

(3.10)

Taking a closer look at the expression for µ(κ) learns that the real part of µ(κ) is equal to orsmaller than 0 for all κ ∈ R if J ≤ 1

εγ . However, for the case that J > 1εγ at least a part of the

µ(κ)-curve, given by −12εα + 1

2

√w+, lies in the right half-plane of the complex plane. Since

this corresponds to values of λ with positive real part (note that the transformation from µto λ only affects the imaginary part), this crossing of the imaginary axis obviously would giverise to an instability of the travelling wave solutions, i.e. a perturbation to the travelling wavesolution which is (initially) close to the solution will not stay close. But, for J > 1

εγ the fixedpoints ϕB

2n = arcsin(εγJ)+2nπ do not exist, so also the travelling wave solutions do not existfor these values of J . Therefore, only values of J smaller than or equal to 1

εγ are relevant andfor these values, the boundary of the essential spectrum lies completely in the left half-plane


June 2005

24 3. Stability

of the complex plane. It consists of the λ1,2,3,4-curves with real part of the form

<(λ(κ)1,3) =

−12εα, 0 < J ≤ J∗, ∀κ ∈ R

−12εα± 1

2

√w+, J∗ < J ≤ 1

εγ, 0 ≤ |κ| < κ∗(J)

−12εα, J∗ < J ≤ 1

εγ, |κ| ≥ κ∗(J)

(3.11)

and

<(λ(κ)2,4) = −12εα, 0 < J <

1εγ

, ∀κ ∈ R, (3.12)

where J∗ is the maximum value of J for which w+ ≤ 0 for all κ. If J > J∗, the minimumvalue of |κ| for which w+ ≤ 0 is denoted by κ∗(J). The imaginary part of the λ1,2,3,4-curvesis of the form

=(λ(κ)1,3) =

±12

√−w+ +

cκ√1− c2

, 0 < J ≤ J∗, ∀κ ∈ Rcκ√

1− c2J∗ < J ≤ 1

εγ, 0 ≤ |κ| < κ∗(J)

±12

√−w+ +

cκ√1− c2

, J∗ < J ≤ 1εγ

, |κ| ≥ κ∗(J)

(3.13)

and

=(λ(κ)2,4) = ±12

√−w− +

cκ√1− c2

, 0 < J ≤ 1εγ

, ∀κ ∈ R. (3.14)

So, solutions converging to the equilibria ϕA,B2n can be stable, but that can not be concluded

before the point spectrum is analyzed. On the other hand, solutions converging to the equi-libria ϕA,B

2n+1 (cos(ϕA2n+1) = −

√1− ε2γ2, cos(ϕB

2n+1) = − 1J

√1− ε2γ2J2) can not be stable,

since now

w+ = ε2α2 + 2

√(1J

√1− ε2γ2J2 −

√1− ε2γ2

)2+ 4

(εκ2S1−c2

)2

− 2(

2κ2

1−c2−

√1− ε2γ2 − 1

J

√1− ε2γ2J2

)

κ=0= ε2α2 + 2

∣∣∣ 1J

√1− ε2γ2J2 −

√1− ε2γ2

∣∣∣ + 2√

1− ε2γ2 + 2 1J

√1− ε2γ2J2

= ε2α2 + 4 max(√

1− ε2γ2, 1J

√1− ε2γ2J2

)> ε2α2,

(3.15)

and hence, −12εα + 1

2

√w+ > 0. This shows that (at least) for κ = 0 a part of the essential

spectrum lies in the right half-plane of the complex plane and hence solutions converging tothese equilibria are always unstable.

3.3 The Point Spectrum

As mentioned before, the spectrum of the operator T (λ) consists of the essential spectrumand the point spectrum which are disjunct. In contrast to the essential spectrum (whichconsists of values of λ such that vA,B(ξ) are oscillating solutions of the eigenvalue problem),the point spectrum Σp consist of values of λ such that V (xi) is a non-oscillating solution of(3.3) which also satisfies the boundary conditions

limξ→±∞

V (ξ) = 0, (3.16)


3.3. The Point Spectrum 25

(a) λ(κ)1 and λ(κ)3 in the complexplane

(b) λ(κ)2 and λ(κ)4 in the complexplane

Figure 3.1 The boundary of the essential spectrum for ε = 0.1, γ = 1, α = 1, J = 2, S = −2and κ ∈ R.

i.e. those λ for which there exists (at least) twice differentiable functions vA,B(ξ) on Rwhich satisfy the boundary conditions limξ→±∞ vA,B(ξ) = 0 and limξ→±∞(vA,B)ξ(ξ) = 0,such that the eigenfunction V (ξ) = [vA(ξ), (vA)ξ(ξ), vB(ξ), (vB)ξ(ξ)]T solves the eigenvalueproblem (3.3). Since Aε(ξ, λ) is the constant operator (Aε)∞(λ) in the limit ξ → ±∞, thesolution V (ξ) to the spectral problem will be an intersection of the unstable manifold of theequilibrium lim

ξ→−∞V (ξ) = 0 and the stable manifold of the equilibrium lim

ξ→∞V (ξ) = 0. So,

the point spectrum consist of those λ for which these unstable and stable manifold, spannedby the corresponding eigenfunctions of (3.3), intersect. These λ are determined by the Evansfunction (D(λ)),

D(λ) = e−∫ ξ0 tr(A(s,λ)) ds(U+

1 ∧ ... ∧ U+n ∧ U−

1 ∧ ... ∧ U−m)(ξ, λ). (3.17)

Here, U+1 , ..., U+

n are n independent eigenfunctions, corresponding to n eigenvalues withnegative real part of the system at ξ = +∞, i.e. U+

1 , ..., U+n satisfy the boundary con-

dition limξ→∞

U+i = 0, i = 1, ..., n. Likewise, U−

1 , ..., U−m are m independent eigenfunctions,

corresponding to m eigenvalues with positive real part of the system at ξ = −∞, i.e.U−

1 , ..., U−j satisfy the boundary condition lim

ξ→−∞U−

j = 0, j = 1, ..., m. Furthermore, (U+1 ∧...∧

U−m)(ξ, λ) stands for the determinant of the matrix who’s columns are given by the functions

U+1 (ξ, λ), ..., U+

n (ξ, λ), U−1 (ξ, λ), ...U−

m(ξ, λ). If for a particular value of λ the Evans functionequals 0, this implies that for that λ the eigenfunctions do not form a linearly independentsystem anymore. The stable manifold spanned by the eigenfunctions bounded on +∞ inter-sects the unstable manifold spanned by eigenfunctions bounded on −∞. Hence, for that λthere exists a solution V (ξ) to problem (3.3) which satisfying both boundary conditions.


June 2005

26 3. Stability

3.3.1 Σp for ε = 0

Consider the linearized system without perturbation terms, i.e. ε = 0. The eigenvalueproblem is then given by

Vξ(ξ) = A0(ξ, λ)V (ξ) =[

AA0 (ξ, λ) O2

O2 AB0 (ξ, λ)

]V (ξ). (3.18)

Hence, the equations for vA(ξ) and vB(ξ) are decoupled and therefore, the system can beanalyzed by investigating the two decoupled systems (VA)ξ(ξ, λ, 0) = AA

0 (ξ, λ)VA(ξ, λ, 0) and(VB)ξ(ξ, λ, 0) = AB

0 (ξ, λ)VB(ξ, λ, 0) separately under the conditions limξ→±∞ VA,B(ξ, λ, 0) =0. At ξ = ±∞, the operator AA

0 (ξ, λ) is given by

limξ→±∞

AA0 (ξ, λ) ≡ (AA

0 )∞(λ) =

[0 1

λ2 + 1 −2cλ√1−c2

]. (3.19)

The eigenvalues κA1,2 of (AA

0 )∞(λ) are of the form σA1 ± σA

2 , where σA1 = −cλ/

√1− c2 and

σA2 =

√λ2 + 1− c2/

√1− c2. Let ΛA

0 be defined as C \ { z | <(z) = 0 ∧ |=(z)| ≥ √1− c2 }.

Then for λ ∈ ΛA0 , the operator (AA

0 )∞(λ) has one eigenvalue (κA1 ) with positive real part

and one eigenvalue (κA2 ) with negative real part. In order to find a solution of the decoupled

system of (3.18) for VA(ξ, λ, 0) which also satisfies both boundary conditions, the values ofλ should be determined for which the stable and the unstable manifold intersect. Let thestable and unstable manifold be given by the exponential decaying eigenfunctions V ±

A (ξ, λ, 0)for ξ → ±∞, respectively. These eigenfunctions are defined in ([7],[16]) and given by

V +A (ξ, λ, 0) =

e(σA1 −σA

2 )ξ

σA2 + 1

[σA

2 + tanh(ξ)(σA

1 − σA2 )(σA

2 + tanh(ξ)) + sech2(ξ)

](3.20)

and

V −A (ξ, λ, 0) =

e(σA1 +σA

2 )ξ

σA2 + 1

[σA

2 − tanh(ξ)(σA

1 + σA2 )(σA

2 − tanh(ξ))− sech2(ξ)

]. (3.21)

They are constructed in such a way that at λ = 0 they coincide with the derivative of thetravelling wave solution ϕA

ε (ξ) at ε = 0, since (ϕA0 )ξ(ξ) solves the linearized system at ε = 0.

Besides that, they are normalized such that limξ→±∞ e−κ2,1ξV ±A (ξ, λ, 0) = [1, κ2,1]T . Now,

the Evans functions turns into

DA(λ, 0) = e−∫ ξ0 tr(AA

0 (s,λ)) ds(V +A ∧ V −

A )(ξ, λ, 0), (3.22)

where the exponential in the Evans function cancels out the exponentials in front of theeigenfunctions V ±

A (ξ, λ, 0). Evaluating this expression, the Evans function is explicitly givenby

DA(λ, 0) = 2√

λ2 + 1− c2

√1− c2

λ2

1− c2

1(σA

2 + 1)2. (3.23)

Solving for which λ the Evans function equals 0 results in λ = 0 (with multiplicity 2) andλ = ±i

√1− c2. Here, one of the zeros of the Evans function at λ = 0 is due to the fact that

at λ = 0 both eigenfunctions coincide with the derivative of the travelling wave solution atε = 0, which solves the linearized system. On the contrary, the values ±i

√1− c2 do not lead

to a solution which satisfies both boundary conditions since they do not lie in ΛA0 , i.e. the

eigenvalues of (AA0 )∞(±i

√1− c2) have zero real part and hence the eigenfunctions V ±

A (ξ, λ, 0)



for λ = ±i√

1− c2 are oscillating solutions.

Next, the decoupled system of (3.18) for VB(ξ, λ, 0) is considered. It should be determined ifthere exist solutions of (VB)ξ(ξ, λ, 0) = AB

0 (ξ, λ)VB(ξ, λ, 0) which satisfy limξ→±∞ VB(ξ, λ, 0) =0. Since ϕB

ε (ξ)|ε=0 = 0, the operator AB0 (ξ, λ) is independent of ξ and is given by

AB0 (ξ, λ) = AB

0 (λ) =

[0 1

λ2 + 1J

−2cλ√1−c2

]. (3.24)

The eigenvalues κB1,2(λ) of AB

0 (λ) are of the form σB1 ±σB

2 , where σB1 = −cλ/

√1− c2 and σB

2 =√λ2 + 1−c2

J /√

1− c2. Let ΛB0 be defined as C \ { z | <(z) = 0 ∧ |=(z)| ≥

√1−c2

J }. Then forλ ∈ ΛB

0 the operator AB0 (λ) has one eigenvalue with positive real part (κB

1 ) and one eigenvaluewith negative real part. For λ ∈ (ΛB

0 )C the eigenvalues of AB0 (λ) are purely imaginary. Let the

stable and unstable manifold be given by the exponential decaying eigenfunctions V ±B (ξ, λ, 0)

for ξ → ±∞, respectively. These eigenfunctions are easily defined since AB0 (λ) is independent

of ξ and are given by

V +B (ξ, λ, 0) = e(σB

1 −σB2 )ξ

[1

(σB1 − σB

2 )

](3.25)

and

V −B (ξ, λ, 0) = e(σB

1 +σB2 )ξ

[1

(σB1 + σB

2 )

]. (3.26)

Also these eigenfunctions satisfy the normalization conditions limξ→±∞ e−κB2,1ξV ±

B (ξ, λ, 0) =[1, κB

2,1]T . For this system the Evans functions turns into

DB(λ, 0) = e−∫ ξ0 tr(AB

0 (λ)) ds(V +B ∧ V −

B )(ξ, λ, 0), (3.27)

which is explicitly given by

DB(λ, 0) = 2σB2 =

2√

λ2 + 1−c2

J√1− c2

. (3.28)

Solving which values of λ are zeros of DB(λ, 0)0 results in λ = ±i√

(1− c2)/J . Similar tothe decoupled system for VA(ξ, λ, 0), where the values ±i

√1− c2 do not lead to a solution

that satisfies the boundary conditions, also here the values ±i√

(1− c2)/J do not lead to asolution which satisfies both boundary conditions limξ→±∞ VB(ξ, λ, 0) = 0. This is because±i

√(1− c2)/J ∈ (ΛB

0 )C and hence the eigenfunctions V ±B (ξ, λ, 0) for λ = ±i

√(1− c2)/J are

oscillating solutions. Therefore, for no λ there exists a (non-trivial) solution of (VB)ξ(ξ, λ, 0) =AB

0 VB(ξ, λ, 0) which also satisfies both boundary conditions.

Subsequently, the whole system (3.18) is considered. Finding a solution V (ξ, λ, 0) whichsatisfies (3.18) and also the boundary conditions limξ→±∞ V (ξ, λ, 0) = 0 comes down tofinding for which λ the Evans function

D(λ, 0) = e−∫ ξ0 tr(A0(s,λ)) ds(V +

1 ∧ V +2 ∧ V −

1 ∧ V −2 )(ξ, λ, 0) (3.29)

equals 0. The eigenfunctions V ±1,2(ξ, λ, 0) can be constructed by extending the eigenfunctions

V ±A,B(ξ, λ, 0) of the decoupled systems. Define these extended eigenfunctions as

V ±1 (ξ, λ, 0) =

(V ±A (ξ, λ, 0))1

(V ±A (ξ, λ, 0))2

00

and V ±

2 (ξ, λ, 0) =

00

(V ±B (ξ, λ, 0))1

(V ±B (ξ, λ, 0))2

, (3.30)


June 2005

28 3. Stability

where (·)1 and (·)2 denotes the first and the second component of (·), respectively. Thenfor λ ∈ Λ0 ≡ C \ { z | <(z) = 0 ∧ |=(z)| ≥ min(

√1− c2,

√(1− c2)/J) }, the matrix A0 has

two eigenvalues with positive real part and two with negative real part. With the aboveeigenfunctions, the Evans function is explicitly given by

D(λ, 0) = 4λ2√

λ2 + 1− c2

√λ2 + 1−c2

J

(1− c2)21

(σA2 + 1)2

. (3.31)

Solving for which λ the Evans function equals 0 results in λ = 0, λ = ±i√

1− c2 and±i

√(1− c2)/J . However, only for λ = 0 there exists a solution V (ξ, λ, 0) which solves (3.18)

and also satisfies both boundary conditions. The other four values of λ lie at the boundaryof the essential spectrum for ε = 0.

Hence for ε = 0, the point spectrum consists of two zeros of the Evans function at λ = 0,

where the other four zeros of the Evans function at λ = ±i√

1− c2 and λ = ±i√

1−c2

J lie inthe essential spectrum, i.e. the values of λ in Σp have zero (and thus non-positive) real part.Nevertheless, if the system is perturbed it might happen that one of the zeros at λ = 0 movesinto the right half-plane or a zero of the Evans function bifurcates out of the essential spec-trum into the right half-plane and cause an instability. Furthermore, an instability can alsooccur if for a small perturbation a zero of the Evans function bifurcates from +∞. However,for λ large the Evans function (for ε = 0) behaves like

D(λ, 0) = O(λ2) (3.32)

and thus no zero of the Evans function can arise from +∞ for small ε. Therefore, the

behaviour of the Evans function near λ = 0, λ = ±i√

1− c2 and λ = ±i√

1−c2

J for ε 6= 0 ¿ 1should be analyzed in order to see if one of these zeros moves into the right half-plane.

3.3.2 Σp near λ = 0 for ε 6= 0

An instability can occur if the zeros of the Evans function at λ = 0 (for ε = 0) move intothe right half-plane when the system is perturbed. Although it is not likely that an explicitexpression of the Evans function can be found for ε 6= 0, it can be analyzed whether the zerosof the Evans function move into the right half-plane and cause an instability or not. Sincefor ε 6= 0 the derivatives (ϕA

ε )ξ(ξ) and (ϕBε )ξ(ξ) of the travelling wave solutions still solve

the linearized system, there is still a zero of the Evans function at λ = 0 for ε non-zero, i.e.D(0, ε) = 0. Furthermore, the series expansion of the Evans function near λ = 0 (for ε = 0)is given by

D(λ, 0) =1

(1− c2)√

Jλ2 + O(λ4). (3.33)

Hence, the Evans function around λ = 0 behaves as a parabola (as can be seen from the seriesexpansion). Therefore, for a small perturbation (i.e. for ε 6= 0 ¿ 1), the Evans function willbe a shifted parabola. Assume that the derivative of the Evans function with respect to λis positive at λ = 0 (the position of one of the zeros of the Evans function for ε 6= 0). Thismeans that the parabola is shifted downwards and to the left and so the other zero of theD(λ, ε) will be on the left of λ = 0. Hence, one of the two zeros of D(λ, 0) at λ = 0 has movedinto the left half-plane, i.e. has negative real part. So, to show that no instability occurs forthe zeros near λ = 0, the derivative of the Evans function with respect to λ at λ = 0 for smallε (i.e. ∂

∂λD(λ, ε)|λ=0) should be determined. The Evans function is given by

D(λ, ε) = e−∫ ξ0 tr(Aε(s,λ)) ds

((V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2

)(ξ, λ, ε). (3.34)



Here, (V ε)+1,2(ξ, λ, ε) and (V ε)−1,2(ξ, λ, ε) are the exponential decaying eigenfunctions whichsolve the equation V ε

ξ (ξ, λ, ε) = Aε(ξ, λ)V ε(ξ, λ, ε) (they are exponential decaying for ξ → +∞and ξ → −∞, respectively). At λ = 0 the eigenfunctions (V ε)±1 (ξ, 0, ε) are defined by thederivative of the travelling wave solution, i.e.

(V ε)+1 (ξ, 0, ε) = (V ε)−1 (ξ, 0, ε) = (Cε)1

(ϕAε )ξ

(ϕAε )ξξ

(ϕBε )ξ

(ϕBε )ξξ

, (3.35)

where (Cε)1 is a multiplicative constant. Using the fact that (V ε)+1 (ξ, 0, ε) = (V ε)−1 (ξ, 0, ε),the expression of the derivative of the Evans function at λ = 0 turns into

(∂

∂λD

)(0, ε) = e−

∫ ξ0 tr(Aε(s,0)) ds

[((∂

∂λ(V ε)+1

)∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2

)(ξ, 0, ε)

+((

∂

∂λ(V ε)−1

)∧ (V ε)+1 ∧ (V ε)+2 ∧ (V ε)−2

)(ξ, 0, ε)

]= G+(ξ, 0, ε) + G−(ξ, 0, ε).

(3.36)

For further examination of the expression for the derivative of the Evans function, the follow-ing Lemma is used.

Lemma 3.1 Let the equation(

∂

∂ξV

)(ξ, λ) = A(ξ, λ)V (ξ, λ)

have four solutions V1(ξ, λ), V2(ξ, λ), V3(ξ, λ) and V4(ξ, λ), for which it holds that limξ→∞

G(ξ, λ) =

G+(λ) exists, where

G(ξ, λ) = e−∫ ξ0 tr(A(s,λ)) ds

((∂

∂λV1

)∧ V2 ∧ V3 ∧ V4

)(ξ, λ).

Then G(ξ, λ) can be rewritten to

G(ξ, λ) = G+(λ) +∫ ξ

∞e−

∫ τ0 tr(A(s,λ)) ds

((∂

∂λA

)V1 ∧ V2 ∧ V3 ∧ V4

)(τ, λ) dτ.

Proof See Appendix B, page 83. ¤

RemarkThe Lemma can also be used for the situation that limξ→−∞G(ξ, λ) exists andis given by G−(λ). The rewritten form of G(ξ, λ) is similar to the one of the Lemma, butwith G+(λ) replaced by G−(λ) and the integral over τ from ∞ to ξ replaced by an integralover τ from −∞ to ξ.

Using the Lemma (where limξ→±∞

G±(ξ, 0, ε) = 0), the expression for the derivative of the

Evans function can be rewritten to(

∂

∂λD

)(0, ε) =

∫ −∞

∞e−

∫ τ0 tr(Aε(s,0)) ds

((∂

∂λAε

)(V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2

)(τ, 0, ε) dτ.

(3.37)

Although the eigenfunctions (V ε)±1 (ξ, 0, ε) are already defined, the other two eigenfunctionsV ±

2 (ξ, 0, ε) are still unknown.


June 2005

30 3. Stability

So, let (V ε)±2 (ξ, 0, ε) be of the form (Cε)±2 [(vεA)±2 , ((vε

A)±2 )ξ, (vεB)±2 , ((vε

A)±2 )ξ]T . Then by substi-tuting the series expansion in ε of (V ε)±2 (ξ, 0, ε) in V ε

ξ = AεV , the expression of (V ε)±2 (ξ, 0, ε)can be determined up to O(ε). These eigenfunctions are determined by using the fact that theexpressions for ε = 0 should match the eigenfunctions V ±

2 (ξ, λ, 0) of the previous section atλ = 0, i.e. (V ε)±2 (ξ, 0, ε)

∣∣ε=0

= V ±2 (ξ, λ, 0)

∣∣λ=0

. Furthermore, they should satisfy the bound-ary conditions limξ→±∞(V ε)±2 (ξ, 0, ε) = 0 for ε 6= 0. If now the integrand in the derivative ofthe Evans functions is also expanded in a series of ε, this leads to(

∂

∂λD

)(0, ε) =

εα

(1− c2)√

J+ O(ε2). (3.38)

Since α > 0 and |c| < 1 this shows that for ε 6= 0 the derivative of the Evans function withrespect to λ at λ = 0 is indeed positive. So, one of the zeros of D(λ, 0) at λ = 0 has movedinto the left half-plane, as explained at the beginning of this section, and hence causes noinstability.

3.3.3 Σp near λ = ±i√

1− c2, ±i√

1−c2

Jfor ε 6= 0

An instability can also arise from a zero of the Evans function at λ = ±i√

1− c2 or λ =

±i√

1−c2

J which moves into the right half-plane, if the system is perturbed. These zeros ofthe Evans function lie on the boundary of the essential spectrum. To be more precise, theylie at the points for which the operator (A0)∞(λ) has a double eigenvalue. This can be seenfrom the expressions of the essential spectrum (expressions (3.11, 3.12, 3.13, 3.14) togetherwith (3.10)). Since the expressions for Σess have κ (which corresponds to the eigenvalues of(Aε)∞(λ)) as parameter, a λ belonging to one of the curves corresponds to two different κ’s.If now the (absolute value of the) imaginary part of λ decreases, the two eigenvalues for thatλ will approach each other until they coincide. They coincide for that λ on the curve withminimal imaginary part. Hence, a λ for which the operator (Aε)∞(λ) has a double eigenvaluecorresponds to an end point of the essential spectrum. For ε = 0, e.g., the end points of thecurves of the essential spectrum lie at λ = ±i

√1− c2 and λ = ±i

√(1− c2)/J . Therefore, an

instability can only occur if a zero of the Evans function bifurcates from an end point of theessential spectrum into the right half-plane.

For the perturbed system, the characteristic polynomial of the operator (Aε)∞(λ) is given by

p(Aε)∞(λ)(κ) =(

κ2 − η(λ)1− ν

κ− 12

ω∞(λ) + χ∞(λ)1− ν

)×

(κ2 − η(λ)

1 + νκ− 1

2ω∞(λ) + χ∞(λ)

1 + ν

)− 1

4(ω∞(λ)− χ∞(λ))2

1− ν2,

(3.39)

whereω∞(λ) = λ2 + εαλ +

√1− ε2γ2,

χ∞(λ) = λ2 + εαλ + 1J

√1− ε2γ2J2

η(λ) =−2cλ− εαc√

1− c2and ν = ε

S

1− c2.

(3.40)

Equal Junctions

For J = 1, the explicit expressions of λ for which (Aε)∞(λ) has a double eigenvalue can bedetermined and are given by

λ = −12εα±(1) i

√1− c2

√1±(2) ν

1±(2) ν(1− c2)

(√1− ε2γ2 − 1

4ε2α2

), (3.41)



where all four possible combinations of ±(1) and ±(2) for λ give a double eigenvalue. Hencefor ε 6= 0, these points move into the left half-plane at a speed of O(ε). Furthermore, forε = 0 the λ’s with ±(1) a plus are the end points of the essential spectrum at i

√1− c2 and

the λ’s with ±(1) a minus are the end points at −i√

1− c2. From here on, the Evans functionwill be analyzed for the end point near i

√1− c2 with ±(2) a minus (the analysis near the

other end points is similar). A new parameter ς is defined as follows,

ς = λ +12εα− λ(ε), (3.42)

where

λ(ε) = i√

1− c2

(√1− ν

1− ν(1− c2)

(√1− ε2γ2 − 1

4ε2α2

)− 1

). (3.43)

This new parameter is used to make the transformations

Aε(ξ, ς) = Aε(ξ, λ) and D(ς, ε) = D(λ, ε), so (Aε)∞(ς) = (Aε)∞(λ). (3.44)

As a result of these transformations, the end point of the essential spectrum (i.e. the pointfor which (Aε)∞(ς) has a double eigenvalue) occurs at ς0 = i

√1− c2, i.e. a fixed value for ς

which is independent of ε. This eigenvalue of (Aε)∞(ς0) of algebraic multiplicity 2 is given by

κ = − ic

1− ν

√1− ν

1− ν(1− c2)

(√1− ε2γ2 − 1

4ε2α2

). (3.45)

Now, to determine if the zero of the Evans function at i√

1− c2 for ε = 0 bifurcates out ofthe essential spectrum into the right half-plane when the system is perturbed, the derivativeof the Evans function with respect to ε at that end point should be evaluated. This meansfinding the expression for

(∂∂εD

)(ς0, 0),

(∂

∂εD

)(ς0, 0) =

∂

∂ε

(e−

∫ ξ0 tr(Aε(s,ς)) ds

(V +

1 ∧ V +2 ∧ V −

1 ∧ V −2

))(ξ, ς, ε)

∣∣∣∣(ξ,ς,ε)=(ξ,ς0,0)

(3.46)

where V ±1,2(ξ, ς, ε) solve the equation Vξ(ξ, ς, ε) = Aε(ξ, ς)V (ξ, ς, ε). For ε = 0, it holds

that λ = ς = ς0. Therefore, the functions V ±1,2(ξ, ς, ε) at (ξ, ς, ε) = (ξ, ς0, 0) are given by

the eigenfunctions V ±1,2(ξ, λ, 0)|λ=ς0 which are determined for the analysis of the point spec-

trum of the unperturbed system (see Section 3.3.1). Hence, for λ = ς0 = i√

1− c2 followsV +

1 (ξ, ς0, 0) = −V −1 (ξ, ς0, 0) and V +

2 (ξ, ς0, 0) = V −2 (ξ, ς0, 0), so all terms in the derivative of

D(ς, ε) with respect to ε at (ς, ε) = (ς0, 0) vanish, i.e.(

∂

∂εD

)(ς0, 0) = 0, for J = 1. (3.47)

Although this does not rule out that the zero of Evans function bifurcates out of the essentialspectrum, it does show that if the zero of the Evans function bifurcates out of the essentialspectrum it can only move at a velocity of O(ε2). Since the essential spectrum moves intothe left half-plane at a speed of O(ε), this implies that the zero of the Evans function nearthe end point of the essential spectrum for ε 6= 0 also has a negative real part. Hence, noinstability occurs for a small perturbation of a two-fold stack of equal junctions.


June 2005

32 3. Stability

Unequal Junctions

Next, consider J 6= 1. In this case, the end points of the essential spectrum are assumed tolie at

λ = −12εα± i

√1− c2w1(ε, J) and at λ = −1

2εα± i

√1− c2

Jw2(ε, J), (3.48)

for some w1(ε, J) and w2(ε, J) which satisfy w1,2(ε, 1) =√

1∓(2)ν

1∓(2)ν(1−c2)(√

1− ε2γ2 − 14ε2α2)

(here w1(ε, 1) and w2(ε, 1) correspond to the minus and the plus of ∓(2), respectively) andw1(0, J) = w2(0, J) = 1. Hence, for ε = 0 these λ’s are the end points of the essentialspectrum of the unperturbed system and for J = 1 they are the end points of the essentialspectrum of the perturbed system with equal junctions.

The end point near λ = i√

1− c2

Again, the Evans function will be analyzed for the end point near λ = i√

1− c2. For thissituation the parameter ς is given by

ς = λ +12εα− λ(ε, J), where λ(ε, J) = i

√1− c2 (w1(ε, J)− 1) . (3.49)

So, (Aε)∞(ς) = limξ→∞

Aε(ξ, ς − 12εα + λ(ε, J)) has a double eigenvalue at ς = ς0 = i

√1− c2.

For the eigenfunctions V ±1 (ξ, ς, ε) it still holds that for ς = ς0 and ε = 0 they are related

as before, i.e. V +1 (ξ, ς0, 0) = −V −

1 (ξ, ς0, 0). However, the relation V +2 (ξ, ς0, 0) = V −

2 (ξ, ς0, 0)is not valid anymore. Therefore, the expression of the derivative of the transformed Evansfunction with respect to ε at ς = ς0 and ε = 0 is given by

(∂

∂εD

)(ς0, 0) = e−

∫ ξ0 tr(Aε(s,ς)), ds

{((∂

∂εV +

1

)∧ V +

2 ∧ V −1 ∧ V −

2

)(ξ, ς, ε) +

+((

∂

∂εV −

1

)∧ V +

1 ∧ V +2 ∧ V −

2

)(ξ, ς, ε)

}∣∣∣∣(ξ,ς,ε)=(ξ,ς0,0)

= G+(ξ, ς0, 0) + G−(ξ, ς0, 0).

(3.50)

This expression can be rewritten by using Lemma 3.1 to

(∂

∂εD

)(ς0, 0) =

=∫ −∞

∞e−

∫ τ0 tr(A0(s,ς0)) ds

((∂

∂εAε

)V +

1 ∧ V +2 ∧ V −

1 ∧ V −2

)(τ, ς0, 0) dτ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0).

(3.51)

Since for ς = ς0 and ε = 0 the eigenfunctions V ±1,2(ξ, ς, ε) are explicitly known as well as the

expression of Aε(ξ, ς), the complete integrand can be evaluated. Hence, the derivative of thetransformed Evans function with respect to ε at the end point of the essential spectrum is



given by

(∂

∂εD

)(ς0, 0) =

∫ −∞

∞2

√1J− 1

tanh(ξ)2 sin(ϕA0 (ξ))ϕA

1 (ξ)︸︷︷︸1st term

+

+(

∂

∂εw1

)(0, J)

(2 tanh(ξ)2 + 2ic tanh(ξ) sech(ξ)2

)︸︷︷︸

2nd term

dξ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0).

(3.52)

(Note that for J = 1 the integrand is zero and G±(ξ, ς0, 0) = 0 holds, which again shows that(∂∂εD

)(ς0, 0) equals 0 for J = 1.) The first term in the integrand is an odd function in ξ,

since tanh(ξ)2 and ϕA1 (ξ) are odd (see Section 2.4.1) whereas sin(ϕA

0 (ξ)) is odd. Hence, thisterm vanishes by integrating ξ over the whole R. However, the second term is the sum of aneven (tanh(ξ)2) and an odd (tanh(ξ) sech(ξ)2) function. Therefore, the resulting expressionof the derivative of the Evans function is

(∂

∂εD

)(ς0, 0) =

∫ −∞

∞4

√1J− 1 tanh(ξ)2

(∂

∂εw1

)(0, J) dξ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0).(3.53)

The end point near λ = i√

(1− c2)/J

The same analysis as the analysis of the Evans function near the end point i√

1− c2 can be

performed near the end point of the essential spectrum at i√

1−c2

J . Hence, the parameter ς,which is used for the transformation of the Evans function, is defined by

ς = λ +12εα− i

√1− c2

J(w2(ε, J)− 1), (3.54)

such that the end point of the essential spectrum for ε ≥ 0 lies at the fixed value ς0 = i√

1−c2

J ,

which is independent of ε. Since now for the eigenfunctions V ±2 (ξ, ς, ε) at ς = ς0 = i

√1−c2

J and

ε = 0 holds that V +2 (ξ, ς0, 0) = V −

2 (ξ, ς0, 0), whereas the relation V +(ξ, ς0, 0) = −V −(ξ, ς0, 0)does not hold anymore, the derivative with respect to ε of the transformed Evans functionevaluated at ε = 0 becomes(

∂

∂εD

)(ς0, 0) = − e−


{((∂

∂εV +

2

)∧ V +

1 ∧ V −1 ∧ V −

2

)(ξ, ς, ε) +

+((

∂

∂εV −

2

)∧ V +

1 ∧ V +2 ∧ V −

1

)(ξ, ς, ε)

}∣∣∣∣(ξ,ς,ε)=(ξ,ς0,0)

= G+(ξ, ς0, 0) + G−(ξ, ς0, 0).

(3.55)

Hence, it is rewritten by using Lemma 3.1 to(

∂

∂εD

)(ς0, 0) =

= −∫ −∞

∞e−


((∂

∂εAε

)V +

2 ∧ V +1 ∧ V −

1 ∧ V −2

)(τ, ς0, 0) dτ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0),

(3.56)


June 2005

34 3. Stability

where evaluation of the integrand (directly) results in(

∂

∂εD

)(ς0, 0) =

∫ −∞

∞

4J2

√1− 1

J

(∂

∂εw2

)(0, J) dξ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0).(3.57)

Since the derivative of the (transformed) Evans function with respect to ε, near the end pointsi√

1− c2 and i√

(1− c2)/J , in both cases a diverging integral contains (note that J 6= 1 andw1,2(ε, J) are not even in ε, which can be seen from w1,2(ε, 1)), it can not be shown that thezeros of the Evans function which (might) bifurcate out of the essential spectrum also stay inthe left half-plane, i.e. for ε 6= 0 and J 6= 1 it is not known for all values in the point spectrumif they have negative real part or not and hence it cannot be concluded if the travelling wavesolutions are (linearly) (un)stable.


35

Chapter 4

Numerical Results

In this chapter are the numerical results discussed, that are obtained for several differentsettings of a two-fold stack in the [1|0] state. First, the Swihart velocities are introduced, whichplay a role in the maximum velocity that the fluxon of the [1|0] state can reach. Subsequently,the obtained numerical results for the c−γ solution branches and the corresponding solutionsto the differential system are treated.

4.1 The Swihart Velocities

The dynamics of the perturbed system are governed by the equations

ϕAξξ −

S

1− c2ϕB

ξξ = sin(ϕA)− αc√1− c2

ϕAξ − γ

︸︷︷︸≡JA

ϕBξξ −

S

1− c2ϕA

ξξ =sin(ϕB)

J− αc√

1− c2ϕB

ξ − γ

︸︷︷︸≡JB

.(4.1)

Rewriting the equations, such that the second order derivatives of ϕA,B(ξ) depend on lowerorder derivatives only, i.e. solving the equations for ϕA

ξξ(ξ) and ϕBξξ(ξ), leads to

ϕAξξ =

(1− c2)2JA + (1− c2)SJB

(1− c2)2 − S2

ϕBξξ =

(1− c2)SJA + (1− c2)2JB

(1− c2)2 − S2

(4.2)

This shows that the equations become singular if (1 − c2)2 equals S2. Define c± =√

1± S,then the equations become singular for the case that |c| = c+ or |c| = c−. The velocitiesc± are called the Swihart velocities, where c+ is the lowest one because of the fact that thecoupling coefficient S is assumed to be negative. To see the significance of these Swihartvelocities, assume the situation that both junctions are completely identical, i.e. J = 1.Consider the case of an anti-symmetric situation. Then there is a fluxon in the first junctionand in the second junction an anti-fluxon, i.e the same fluxon as in the first junction, butwith an opposite polarity. Then the phase differences ϕA and ϕB are equal to each other, butwith opposite sign. The equations now reduce to

ϕAξξ =

(1− c2)(1− c2 − S)JA

(1− c2 + S)(1− c2 − S)=

(1− c2)JA

1− c2 + S

ϕBξξ =

(1− c2)(1− c2 − S)JB

(1− c2 + S)(1− c2 − S)=

(1− c2)JB

1− c2 + S

(4.3)


June 2005

36 4. Numerical Results

which both are scaled equations of a single junction. Hence, scaling the travelling wave coor-dinate to x−ct√

1−c2+S, such that the equations are again of the form ϕA,B

ξξ = JA,B (with√

1− c2

replaced by√

1− c2 + S) shows that the maximum velocity for the anti-symmetric situation isgiven by |c| = √

1 + S, i.e. the lowest Swihart velocity. For the symmetric case, i.e. both junc-tions contain a fluxon with the same polarity, the equations reduce to ϕA,B

ξξ = (1−c2)(1−c2−S)

JA,B.

For this situation the maximum velocity is given by |c| =√

1− S, i.e. the highest Swihartvelocity.

Next, consider the coupled junctions in the [1|0] state. Suppose that the velocity of the fluxonin the first junction reaches the lowest Swihart velocity, i.e. it moves with (absolute) velocity|c| = √

1 + S. By substituting this Swihart velocity for the velocity in the equations of (4.1)the relation JA − JB = 0 is obtained, or more explicitly

sin(ϕA) =1J

sin(ϕB)− α√

1 + S√−S(ϕB

ξ − ϕAξ ). (4.4)

Because of the presence of a fluxon in the first junction, it holds that ϕA shows an increaseof 2π as ξ changes from −∞ to ∞. Therefore, sin(ϕA) runs over all values from −1 to 1. Ifnow J equals 1 and α = 0, the increase of ϕA of 2π implies that ϕB must increase similar,i.e. there is also a fluxon in the second junction. Since this on its turn implies a [1|1] state,there is a contradiction to the assumption that a [1|0] state is considered. Hence it followsthat for J = 1 the velocity of the fluxon of the [1|0] state can not reach the lowest Swihartvelocity c+.Likewise for J > 1 and α = 0, a moving fluxon of the [1|0] state can not reach a velocityof c+. This can also be seen from relation (4.4). Assume that the fluxon would move withvelocity c+, then from ϕA = π/2 it follows that sin(ϕB) = J . Since J was assumed to belarger than 1 there is no phase difference ϕB which can satisfy sin(ϕB) = J .At last, for J < 1 (and α = 0) it is possible for a fluxon in the [1|0] state to move at the lowestSwihart velocity. An increase of ϕA to π/2 leads to an increase of ϕB to arcsin(J) < π/2.If now ϕA increases more from π/2, sin(ϕA) starts decreasing and hence sin(ϕB) also startsdecreasing. Since ϕB was still less than π/2, this implicates that ϕB decreases from arcsin(J).So, if ϕA increases by 2π as ξ changes from −∞ to ∞, ϕB will not experience a 2π increase,i.e. for J < 1 a fluxon in the first junction can move with a velocity equal to the lowestSwihart velocity whereas there is no fluxon in the second junction and hence the [1|0] stateis sustained.

IV-characteristic

The relation between the applied bias current density and the velocity of the travelling wavesolution (γ − c-curve) is referred to as the IV -characteristic, since this is how the relationis measured in practice. If a bias current I is applied to a junction (containing a fluxon),the velocity of the (moving) fluxon is changed. Due to applying a bias current, there will bea potential difference over the junction resulting in the ac-Josephson effect, i.e. the phasedifference will change in time due to the potential difference. The ac-Josephson relation of

this effect is given by ([16]) ∂tϕ =2e

~V , where ϕ is the phase difference in the junction and

V is the potential difference over the junction. Since the fluxon corresponds to an increase ofthe phase difference of 2π, the total voltage over the junction (of length L) is given by

∫

L

V dx =~2e

∫

L

ϕt dx =h

2π 2e

∫

L

−cϕx dx = −ch

2e, (4.5)


4.2. Unequal Junctions (J > 1) 37

if there is a fluxon moving with velocity c. Therefore, the total voltage over the junction isproportional to the velocity of the fluxon, whereas this velocity in turn depends on the biascurrent that is applied. Since the bias current which is applied is known, by measuring thetotal voltage over the junction the relation between I and V is obtained (IV-characteristic)and hence the relation between γ and c is known.

−1 −0.8 −0.6 −0.4 −0.2 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

c

γ

A

B

Figure 4.1 The IV-characteristic for J = 2, S = −0.2 and α = 0.18.

4.2 Unequal Junctions (J > 1)

Next, the numerical results are presented. In this and the following sections, the IV-characteristicsfor different values of the coupling parameter S for the different situations of (1) unequaljunctions with J > 1 (this section), (2) equal junctions (J = 1, Section 4.3) and (3) unequaljunctions with J < 1 (Section 4.4). As said, here the situation is treated that the junctionsare not identical to each other, where J is considered to be larger than 1, i.e. the criticalcurrent density of the first junction is larger than that of the second junction. The numericalresults are obtained by using the AUTO software package ([8]). In Appendix C is the usedAUTO program demonstrated, which is used to obtain (parts of) the IV-characteristic for thespecific situation that J = 2 and the values −0.1456 and −0.15 of the coupling parameter S.

4.2.1 Backbending

An interesting phenomenon that occurs for J larger than 1 is “backbending”. This backbend-ing is visible in the IV-characteristic in Figure 4.1. Starting on the left part of the branch,the velocity of the fluxon increases if the applied bias current increases. However, at a cer-tain point (before the top of the curve) the velocity starts decreasing while the applied biascurrent still increases, i.e. the IV-curve bends back. The backbending phenomenon is due toan increase of the gradients in the junctions (see Figure 4.2). For α 6= 0, the dissipation inthe junctions is proportional to the velocity times the gradients (recall the −αc√

1−c2ϕA,B

ξ terms).If γ changes, as a result the fluxon will move with a different velocity, such that for the newvelocity the energy input (γ) and the energy dissipation (the α term) are balanced. For asmall value of γ (and hence for small c) the gradients in the junctions are (relatively) small,so an increase of γ will speed up the fluxon. Under an increase of the γ, the so-called imagein the second junction (due to the coupling) of the fluxon in the first junction grows and thusgrows the gradient in the second junction. Furthermore, the fluxon in the first junction also


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−20 −15 −10 −5 0 5 10 15 20−0.5

0

0.5

1

1.5

2

2.5

3

ξ

φAξ

(a) derivative of the solution in the first junction forγ = 0.1 (solid curve) and γ = 0.27 (dashed curve)

−20 −15 −10 −5 0 5 10 15 20

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

ξ

φBξ

(b) derivative of the solution in the second junctionfor γ = 0.1 (solid curve) and γ = 0.27 (dashed curve)

Figure 4.2 The derivatives of two solutions on the left part of the branch of Figure 4.1.

contracts as a result of moving with a larger velocity, i.e. also in the first junction the gradientincreases. If now the bias current keeps increasing, at a certain point the gradients grow thatfast that by further increasing γ the velocity has to decrease in order to balance the energydissipation and the energy input. So observing Figure 4.1, the backbending phenomenonoccurs for those values of the bias current between the value of γ for which the velocity ismaximal (i.e. γ for the most left point of the curve) and the maximum value of γ for whichthe solution exists (i.e. γ at the top of the curve).

Besides the increase of the gradients in the junctions, also from Figure 4.2 the difference inshape of the derivative of the solution in the second junction for increasing γ is observable.For a small applied bias current (γ = 0.1), the gradient in the second is symmetric aroundξ = 0, which is in agreement with the derived analytical solution in the second junctionof order ε (see Section 2.4.2 and (ϕB

1 )ξ(ξ) in Figure 2.5). For a larger applied bias current(γ = 0.27) this is not the case anymore. It is clearly visible in Figure 4.2 that for γ = 0.27the maximum value of ϕB

ξ (ξ) for ξ negative is larger than the maximum value for ξ positive.Hence, it may be concluded that the perturbation analysis for a small applied bias current(at least up to γ = 0.1) results in a good approximation of the numerical computation.

Along the right part of the branch in Figure 4.1, decreasing the current causes the image in thesecond junction to grow further to a large hump whereas the velocity further decreases due tothe larger gradient. The hump approaches the height of 2π and then begins to broaden. Nowfor γ approaching 0, the second junction contains a fluxon/anti-fluxon pair, as observable inFigure 4.3. This means that for γ approaching 0 on the right part of the branch the systemno longer is in the [1|0] state, but in fact in a [1|1,−1] state. The anti-fluxon in the secondjunction moves together with the fluxon in the first junction at the same position. The fluxonin the second junction moves at a distance from the anti-fluxon. Since the fluxon and theanti-fluxon have an opposite polarity (i.e. they rotate in opposite directions) the attract eachother. On the other hand, they are pushed away from each other due to Lorentz force actingon them (in different directions due to the different rotation). The distance between thefluxon and the anti-fluxon is then such that the attracting and repelling force are balanced.Since the Lorentz force is induced by the applied bias current, for decreasing γ the Lorentzforce will be less and hence increases the distance between the fluxon and anti-fluxon in order



−50 0 50

0

1

2

3

4

5

6

7

ξ

φA,B

(a) solution in the first (solid curve) and second(dashed curve) junction for γ = γs ¿ 1

−50 0 50

0

1

2

3

4

5

6

7

ξ

φA,B

(b) solution in the first (solid curve) and second(dashed curve) junction for γ < γs

Figure 4.3 Two solutions on the right part of the branch (in region B) of Figure 4.1.

to maintain the balance between the forces. In the limiting situation that γ = 0, the two-foldstack will be in the [1| − 1] state, since the fluxon in the second junction must be at thatlarge distance from the anti-fluxon such that the attracting force is negligible. This is becausefor zero applied bias current there is no repelling Lorentz force and hence this (moved away)fluxon can also not influence the fluxon in the first junction. In Figures 4.3(a) and (b) is alsothe effect in the first junction of the anti-fluxon in the second junction observable, i.e. at theposition of the anti-fluxon in the second junction appears the image of this anti-fluxon in thefirst junction.

In ([16, Chapter 5]), it is shown that the solutions on the right part of the branch (i.e. thepart of the branch contained in the regions A and B) are unstable. For the situation that thestack is in the [1|1,−1] state, this can be seen from the physical point of view that the balancebetween the attracting and repelling force is a fragile one. The attracting force depends onthe distance between the fluxon and the anti-fluxon, while the repelling force depends on thebias current. Hence, if the distance between the fluxons is slightly changed one of the twoforces will become larger than the other one and the fluxons will approach each other andcollide or move apart from each other.

Subsequently, from Figure 4.1 it is observable that the lowest Swihart velocity is not reached,as was reasoned at the beginning of this chapter. For S = −0.2, as in the Figure 4.1, the valueof the lowest Swihart velocity (c+ =

√1 + S) is approximately 0.894 where the (absolute)

highest velocity reached in the figure is approximately 0.844. Moreover, the top of the IV-curvelies below γ = 1

2 (at approximately γ = 0.436) since the fixed points ϕB2n = arcsin(γJ) + 2nπ

only exist for γJ ≤ 1 (note that J = 2 in Figure 4.1). If S starts increasing towards 0, the topof the curve starts moving upwards whereas the upper part of the curve also starts movingto the left. The upward move occurs since the maximum current which can be applied willapproach γ = 1

J if S approaches 0. The move to the left is the result of the increasing (lowest)Swihart velocity if S increases, i.e. the maximum velocity which can be reached will be higher.

However, that is not the only change which occurs in the solution branch for increasing S.By decreasing the coupling (i.e. increasing S) for a solution belonging to the part of thebranch contained in region A (which is depicted in Figure 4.1) from S = −0.2 to S = −0.14,


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−1 −0.8 −0.6 −0.4 −0.2 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

c

γ

A

B


a new solution for the changed coupling is computed. Subsequently, the IV-curve on whichthis new solution lies can be computed. By following the same procedure for a solutionbelonging to the branch contained in region B (which is also depicted in Figure 4.1), the IV-characteristic for the decreased coupling S = −0.14 is obtained, which is significantly differentfrom the IV-characteristic for S = −0.2 (i.e. the IV-characteristic in Figure 4.1). This newIV-characteristic for S = −0.14 is shown in Figure 4.4. It consists of two separate branchesinstead of the closed loop which was obtained before for a larger coupling. From Figures4.5(a) and (b), it is better observable how the closed loop transforms into these two separatebranches. Starting from the situation that the IV-characteristic consists of two branches (asin Figure 4.4), by increasing the coupling (i.e. decreasing S from S = −0.14), the spiraledpart of the branch of Figure 4.4 stretches out as well as the loop in lower part of this samefigure. This is observable from comparing Figures 4.4 and 4.5(a), where the coupling wasincreased from −0.14 (Figure 4.4) to −0.1456 (Figure 4.5(a)). If the coupling is increasedfurther, for a particular value of S (say S∗) the two branches will meet and create an inter-section point of the two branches. Further increasing the coupling causes the branches atthe intersection point to split up again. However, this time the parts of the branches on theright of the intersection point stay connected as well as the parts of the branches on the leftof the intersection point. These new formed branches then move away from each other if thecoupling is increased. This shows that for J > 1 and a coupling larger than the specific valueS∗ (i.e. S < S∗), e.g. as for Figure 4.1, apart from the closed loop shown in that figure thereis also another branch of solutions inside the loop which ends in a spiral at the one end andgoes to γ and c equal to zero at the other end. Therefore, the IV-characteristic of Figure 4.1is not complete, since it misses (at least) the spiraled branch inside the loop.

The solutions along the closed loop of Figure 4.1 loop are already characterized. On the leftpart of the loop the image in the second junction grows and the 2π-kink (the fluxon) in thefirst junction contracts. Again consider region B as depicted in Figure 4.1. Then this region(roughly) corresponds to the regions B of Figures 4.4 and 4.5(a), which contain the loop onthe lower right part of the IV-characteristic. The region B in Figure 4.5(b), containing partsof the two separate (most right) branches of the IV-characteristics, (roughly) corresponds tothe region B of Figure 4.1 which contains the lower right part of the closed loop. Now, thepart of the branch in Figure 4.1 which is contained in region B corresponds to the right partof the loop contained in corresponding regions B in Figures 4.4 and 4.5(a) and it correspondsto the most right branch of Figure 4.5(b) which is contained in region B. On these parts of



−1 −0.8 −0.6 −0.4 −0.2 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

c

γ

A

B

(a) S = −0.1456

−1 −0.8 −0.6 −0.4 −0.2 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

c

γ

A

B

(b) S = −0.15

Figure 4.5 IV-characteristics for J = 2 and α = 0.18.

the branches the image in the second junction grows into a hump which broadens and formsa fluxon/anti-fluxon pair for γ approaching 0. On the left part of the loops contained in theregions B of Figures 4.4 and 4.5(a) and the left branch contained in the region B of Figure4.5(b), the image in the second junction also transforms into a fluxon/anti-fluxon pair whenγ approaches 0, in a similar way as on the right part of the loops (or the right branch, inthe case of S = −0.15). However, there is a difference with the [1|1,−1] state on the rightpart of the loops (or right branch), which manifests itself in the position of the fluxon andanti-fluxon. On the left part of the loops (or the left branch) the anti-fluxon in the secondjunction is (still) on the right of the fluxon in the second junction (as was also the case on theright part of the loops or the right branch), but now the fluxon in the first and the fluxon inthe second junction move at the same position, instead of the fluxon in the first junction andthe anti-fluxon in the second junction. Again, the distance between the fluxon and anti-fluxonin the second junction is determined by the balance between the attracting and repelling force.

The spirals in Figures 4.4, 4.5(a) and 4.5(b) are contained in the regions A, which (roughly)correspond to the region A of Figure 4.1. Along the most right part of the spirals of Figures4.4 and 4.5(a) (and the most right branch contained in region A of Figure 4.5(b)), the imagein the second junction is increasing. However, before the increased image forms a (2π) hump,the solution branch starts spiraling. Along the spiral (following the inward direction), theimage in the second junction “splits” in two peaks. In Figure 4.6, three solutions in the secondjunction for different parameter values of (γ, c) are shown. The dotted, dashed-dotted andsolid curve in the figure are solutions with parameter values (γ, c) along the spiral in such anorder that, by following the inward direction along the spiral, first the parameter values (γ, c)of the dotted curve are reached, then the (γ, c) of the dashed dotted curve and subsequentlythose of the solid curve. The position of the left peak of the solutions along the spiral remainsfixed and is at the same position of the fluxon in the first junction. The right peak (whichhas an oscillating tail) of the solutions along the spirals moves to the right, away from the leftpeak. The height of the connection between the left peak and the right peak (with its oscil-lating tail) is π−arcsin(γJ), whereas the level of the solutions on the left of the left peak andon the right of the right peak is arcsin(γJ). Figure 4.6 shows that the “closer” the parametervalues (γ, c) are to the centre of the spiral, the further to the right the right peak (with itsoscillating is) has moved (here, closer to the centre means that the parameter values lie moreinward along the spiral, by following in the inward direction). Because of the observation in


June 2005


−50 0 500.5

1

1.5

2

2.5

3

3.5

ξ

φB

Figure 4.6 Solutions in the second junction along the spiral of the IV-characteristic forJ = 2, S = −0.14 and α = 0.18.

Figure 4.6, that the right peak with its oscillating tail moves to the right as the parametervalues (γ, c) approach the centre of the spiral, it is presumably true that at the center of thespiral there is only one peak left (the one at the same position as the fluxon, whereas the rightpeak has moved that far to the right that it can not influence the dynamics near the positionof the fluxon anymore). Hence, there exists a heteroclinic solution connecting the fixed pointsϕB

0 = arcsin(γJ) and ϕB1 = π− arcsin(γJ) for the parameter values (γ, c) at the centre of the

spiral. That the heteroclinic connection only exists for the specific values of (γ, c) at the centreof the spiral can be seen from a dimensional analysis of the stable and unstable subspaces forthose specific values. The solution of equations (4.1) for a certain set of values γ, c, α, S andJ is given by the intersection of the stable subspace of the fixed point φ∞ at ξ = ∞ and theunstable subspace of the fixed point φ−∞ at ξ = −∞. So, φ−∞ is given by (ϕA

0 , 0, ϕB0 , 0) and

φ∞ by (ϕA2 , 0, ϕB

1 , 0) for the situation of the heteroclinic connection. Linearizing equations(4.1) around φ∞ and φ−∞ shows that φ−∞ has a two dimensional unstable subspace for thespecific parameter values at the centre of the spiral. Moreover, also the stable subspace of φ−∞is two dimensional, where all for eigenvalues of φ−∞ have zero imaginary part. Furthermore,the stable subspace of φ∞ for the specific parameter values at the centre of the spiral is onedimensional, where the unstable subspace of φ∞ is three dimensional (φ∞ has one positive realeigenvalue and a complex conjugate pair of eigenvalues with positive real part). Hence, theheteroclinic connection of the fixed points φ−∞ and φ∞ is given by the intersection of the twodimensional unstable subspace and the one dimension stable subspace. However, for param-eter values different from those at the centre of the spiral (where all other parameters valuesremain fixed), the two dimensional unstable subspace of φ−∞ will no longer intersect the onedimension stable subspace of φ∞, since at least three parameter values need to change forthe intersection between the unstable and stable subspace to persist in four dimensional space.

It is already mentioned before that the solutions belonging to the parts of the closed loopcontained in the regions A and B (as is the case for Figures 4.4 and 4.5(b)) are unstable (whichwas shown in ([16])). Hence, it is presumably true that the solutions on the left part of theloop contained in the region B, as in Figures 4.4 or 4.5(a) (or on the left branch contained inthe region B, as in Figure 4.5(b)), are also unstable, since on that part of the loop (or on theleft branch) the solutions show a similar behaviour as on the right part of the loop (or on theright branch), for decreasing γ. Moreover, also the solutions along the spiral are presumably



unstable. An indication that all these solution of the branches contained in the regions A andB of the figures are (presumably) unstable, is given in ([4]). In that article is shown that thetravelling wave solution, corresponding to a single fluxon in a single junction, looses stabilityat the first turning point of the IV-characteristic, i.e. at the top, for the maximum bias currentwhich can be applied. In the article it is numerically shown that for the parameter values(γ, c) of the first turning point of the IV-characteristic, a zero of the Evans function crossesthe imaginary axis and hence moves into the right half-plane. Furthermore, for the parametervalues (γ, c) of every succeeding turning point along the IV-characteristic (i.e. in the casethat there is a spiral present in the IV-characteristic) another zero crosses the imaginary axisand hence stability is never regained.

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

c

γ

(a) J = 1.75

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

c

γ

(b) J = 1.25

Figure 4.7 IV-characteristics for S = −0.1456 and α = 0.18.

4.2.2 No backbending

So far, J has been considered constant (J = 2) and the solution branches were only influencedby changes in the coupling (S). From Figures 4.4, 4.5(a) and (b) was observed that a decreaseof the coupling for the situation of a closed loop with a spiraled branch inside leads to thesituation that the inner and outer branch collide and split up, forming two other separatebranches. However, this same phenomenon occurs if the coupling is fixed and J is altered. Iffor a certain value of J the coupling is large enough (i.e. S < S∗), then the IV-characteristicconsists of a closed loop with a spiral inside. By decreasing J , the inner branch and the rightpart of the outer branch will move to each other until they collide. A further decrease of Jwill cause the branches to split up again, where the branches above the collision point stayconnected to each other and the same holds for the branches below the collision point. InFigure 4.7 is visible that, as a result of decreasing J from 1.75 to 1.25 (for a fixed couplingS = −0.1456), the two branches move away from each other (after the separation has takenplace). Moreover, it is observable that the distance between the lowest point of the spiral(i.e. the turning point near (γ, c) ≈ (0.3,−0.62) in Figure 4.7(a)) and the top of the loop (i.e.the turning point near (γ, c) ≈ (0.2,−0.45) in Figure 4.7(a)) especially increases because thewhole part of the branch which forms the spiral moves upwards. This is due to the fact thatthe maximum bias current for which the solution exists increases as J decreases. Rememberthat the fixed points ϕB

2n = arcsin(γJ) + 2nπ can exist for γJ < 1. Although the maximumbias current which can be applied increases, the maximum (absolute) velocity which is reached


June 2005


still remains below the lowest Swihart velocity c+.

When J is further decreased towards 1, the IV-characteristics shown in Figures 4.8(a) and4.8(b) are obtained (note that in Figure 4.8(a) the curves for J = 1.003 are omitted, for thesake of clearness). What is special about these solution branches is best visible by zoomingin at the top of the branch which contains the spiral, as has been done for Figure 4.8(b)(note that although the peak in Figure 4.8(b) appears to be sharp, in fact the top in thefigure is a smooth curve as can be seen in the enlarged inset in Figure 4.8(b)). Whereas for

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

c

γ

(a) J = 1.001

−0.92 −0.91 −0.9 −0.89 −0.88 −0.870.68

0.7

0.72

0.74

0.76

0.78

0.8

c

γ−0.9188

(b) enlargement of the upper-left part for J = 1.001(solid curve) and J = 1.003 (dashed curve)

Figure 4.8 IV-characteristics for S = −0.1456 and α = 0.18.

J = 1.003 there is still a part of the solution branch for which the backbending phenomenonoccurs, for J = 1.001 this part of the branch has become extremely small. Hence, there isstill a small region for which the velocity decreases as the bias current increases. However,if J is even more decreased (but still larger than 1), the backbending phenomenon does notoccur at all anymore, as will be shown in the following section (see Figure 4.12, where thedotted and dashed curves are for values of J larger than 1, but where there is no backbendingphenomenon).


4.3. (Almost) Identical Junctions (J = 1, J − 1 ¿ 1) 45

4.3 (Almost) Identical Junctions (J = 1, J − 1 ¿ 1)

Next, the situation that the two junctions in the stack are identical (or almost identical) isdiscussed, i.e. J = 1 (or J −1 ¿ 1). In previous works (e.g. in [16]), the IV-characteristic fora [1|0] state with identical junctions is computed. The solution branch is reproduced here andis shown in Figures 4.9(a) and 4.9(b). Figure 4.9(a) shows, besides the IV-characteristic, alsothe relation (2.37). This (perturbation) relation was the necessary condition to be satisfied inorder to have a travelling wave solution (for a small perturbation) which was bounded on R+

as well as on R−. The figure shows that for small values of γ, the IV-characteristic and theperturbation relation are indeed in good agreement. It was already observed in the previoussection that the (in Section 2.4.2) derived analytical solution for the travelling wave solutionin the second junction is in good agreement with the numerical solution for γ = 0.1, but notfor γ = 0.27 anymore.In Figure 4.9(b), an enlargement of the upper-left part of the IV-characteristic of Figure4.9(a) is visible. It shows that also for J = 1 the solution branch is partly made up of aspiral, although the spiral is strongly contracted in one direction. Hence, it might not beimmediately clear from the complete picture of the IV-characteristic that there is a spiral.Nevertheless, the solutions for parameter values along the spiral show a resemblant behaviouras solutions for parameter values along the spiral for non-identical junctions, as in the previoussection. Again it indicates that, also for J = 1, for the parameter values (γ, c) at the centre ofthe spiral the solution (in the second junction) is a heteroclinic connection of the fixed pointsϕB

0 and ϕB1 , whereas the first junction contains a fluxon.

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

c

γ

(a) the IV-characteristic (solid curve) and the rela-tion (2.37) (dashed curve)

−0.92 −0.9195 −0.919 −0.9185 −0.9180.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

c

γ

(b) enlargement of the upper-left part of the solutionbranch

Figure 4.9 The IV-characteristic of a stack of identical junctions for S = −0.1456 andα = 0.18.

Comparing the IV-characteristics of Figure 4.9(a) and Figure 4.8(a), the question raises howthe solution branch can change from the curve with a spiral on the left (as in Figure 4.9(a))to the curve with a spiral on the right (as in Figure 4.8(a)). The only difference between theparameter values of the two solution branches is a change of J from 1 (Figure 4.9(a)) to 1.001(Figure 4.8(a)). Moreover, for J = 1.001 there is also the solution branch present (the smallloop in the lower right part of the IV-characteristic) on which the solution goes to a [1|1,−1]state. A solution on this branch for J = 1.001 is taken, i.e. a solution for which the parametervalues (γ, c) belong to the loop in the lower right part of the IV-characteristic of Figure 4.8(a).


June 2005


Next, J is decreased to 1 and the new solution branch to which the solution with the alteredJ belongs is computed. This learns that for J = 1 the solution branch in the form of theloop in the lower right part of the IV-characteristic also exists. Furthermore, the solutionsbelonging to the parameter values on this loop for J = 1 have an equivalent characterizationas solutions on the loop for J > 1, i.e. the [1|1,−1] state is formed for parameter valueson the right and on the left part of the branch. For parameter values on the right part ofthe branch, the anti-fluxon in the second junction moves at the same position as the fluxonin the first junction. The fluxon in the second junction moves away from the anti-fluxon asthe applied bias current decreases. For parameter values on the left part of the branch, thefluxon in the second junction moves at the same position of the fluxon in the first junction.The anti-fluxon in the second junction moves away from fluxon in the second junction as theapplied bias current decreases.Subsequently, this same procedure as for a solution belonging to the small loop is performedfor a solution belonging to the spiral of Figure 4.8(a). For a solution for which its parametervalues (γ, c) belong to the spiral for J = 1.001, J is decreased to 1 and the new solutionbranch to which this altered solution belongs to is computed. The resulting new solutionbranch for J = 1 is a somewhat “headphone” shaped curve which ends on both sides in aspiral. In Figure 4.10(a), the extension of the IV-characteristic of Figure 4.9(a) is shown (note

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

c

γ

(a) the extended picture of solution branches

−0.92 −0.9 −0.88 −0.86 −0.84 −0.82 −0.8

0.55

0.6

0.65

0.7

0.75

0.8

c

γ

(b) enlargement of the upper-left part of the IV-characteristic


that the perturbation relation (2.37) which was visible in Figure 4.9(a), is left out in Figure4.10(a)).

Again, the solutions for parameter values along the left spiral of the headphone curve displaythe same behaviour as the solutions belonging to the spirals which were obtained before.For the parameter values at the centre of the spiral, the solution in the second junction is(most likely) a heteroclinic connection. From Figure 4.10(b) it is also observable that theheadphone curve crosses the solution branch which was primary obtained in Figure 4.9(a).Under a small increase of J , the left part of the headphone branch and the primary branchwill move towards each other, as is shown in Figure 4.11(a). This movement goes on until thetwo branches meet. Just as what happened for the collision between the branches for unequaljunctions, also here the solution branches split up again if J is further increased. The parts ofthe branches above the collision point stay connected and move upwards, whereas the partsof the branches below the collision point stay connected and move downwards. This results



in the shape of the solution branches shown in Figure 4.11(b). Now, the large spiral on the

−0.922 −0.92 −0.918 −0.916 −0.914 −0.9120.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

c

γ

(a) J = 1 (solid curve) and J = 1.00005 (dashedcurve)

−0.922 −0.92 −0.918 −0.916 −0.9140.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

c

γ

(b) J = 1.0005

Figure 4.11 Enlargements of the upper-left part of IV-characteristics for S = −0.1456 andα = 0.18.

right of the primary branch is connected to the primary branch and the two small spiralson the left of the primary branch are connected to each other, forming a separate solutionbranch. Although the large spiral on the right now is connected to the primary branch (whichfollows the perturbation relation for small γ), at the top of the curve there is still a significantdifference with the IV-characteristic shown in Figure 4.8(b). For J = 1.001 the solutionbranch curves to the right at the top, whereas for J = 1.0005 the solution branch curvesto the left at the top and intersects itself. Therefore J is (slowly) further increased to seehow the branch changes from the one shown in Figure 4.11(a) to the other one shown inFigure 4.8(b). In Figure 4.12(a) is shown what happens near the top of the IV-characteristicif J is slowly increased. It is observable that the intersection point moves upwards along theprimary branch, whereas the loop to the left contracts in such a way that it becomes morenarrow. Eventually, the intersection point reaches the top of the curve (where presumably acusp is formed for this particularly value of J) and hence the loop to the left vanishes. If Jis further is decreased, the solution branch curves immediately to the right at the top. Sincethis is also the case for the solution branch which is shown in Figure 4.8(b), the link has beenlaid between the case J = 1 and J > 1 and it is almost completely clear how the solutionbranches change if the junctions in the stack transform from unequal junctions with J > 1 toidentical ones (with J = 1). However, there are still two remarks left to make.

The first remark concerns the intersection the solution branch makes as shown in, e.g., Figure4.11(b). As was observed for J > 1, it can happen that two parts solution branches meeteach other for a particular value of J , creating a collision point. A change in the parameterJ causes the parts of the branches to split up again at the collision point. However, as canbe observed from Figure 4.12(a), the intersection point created by the loop that the solu-tion branch makes near the top persists under a change in J . The intersection point movesupwards along the branch instead of splitting up. This can be understood by observing thesolutions for the parameter values at the intersection point. Following the solutions for pa-rameter values along the branch, the solutions in the second junction for the first and thesecond time that the parameters take the values of the intersection point are shown in Figure


June 2005


−0.919 −0.918 −0.917 −0.916 −0.915 −0.914

0.78

0.785

0.79

0.795

c

γ

(a) enlargement of the IV-characteristics for J =1.0005 (dotted curve), J = 1.0007 (dashed curve) andJ = 1.001 (solid curve)

−20 −15 −10 −5 0 5 10 15 20

0.5

1

1.5

2

2.5

3

ξ

φB

(b) the solutions in the second junction for the first(solid curve) and the second (dashed curve) passthrough the intersection point for J = 1.0005

Figure 4.12 The IV-characteristic and the corresponding solutions in the second junctionfor S = −0.1456 and α = 0.18.

4.12(b). Figure 4.12(b) shows that although the parameter values for both solutions are thesame, the solutions itself are clearly not the same. The image in the second junction keepsincreasing by following the loop of the solution branch, even if the bias current decreases.Furthermore, also in the first junction the solutions are not the same for both times that theparameters take the values of the intersection. However, the difference between these twosolutions is less clearly visible (and therefore not shown in a figure).

The second remark that must be made is the different direction and position of the two leftspirals in Figure 4.11(a). For J = 1 in the figure, the spiral is on the right of the primarybranch and the spiraling direction is counterclockwise, whereas for J = 1.0005 in Figure4.11(a) the spiral is on the left of the primary branch and the spiraling direction is clockwise.In Figure 4.13 is shown what the shape of the spiral undergoes if J increases from 1 to1.00003. As mentioned, for J = 1 the spiral is on the right side of the branch and spiralscounterclockwise. If J is slightly increased, the part of the spiral which comes after the lowestturning point (i.e. the first turning point of the most left spiraled branch in Figure 4.13) movesto the left (it approaches the branch) and crosses the branch, such that this first turning pointnow is a turn to the left. Hence, the spiral is on the left of the branch, but (except for thefirst turning point) still spirals counterclockwise. By slightly further increasing J , the partof the spiral which comes after the second turning point (i.e. the highest turning point ofthe second from the right solution branch in Figure 4.13) moves to the right and crosses thepart of the branch before the second turning point. Hence, also the direction of the secondturning point is reversed, i.e. the first two turning points now are spiraling clockwise, wherethe part of the solution branch after the third turning point still spirals counterclockwise. IfJ is further increased, this procedure of the counterclockwise part of the spiral crossing thesolution branch is repeated infinitely many times such that the spiral becomes a clockwisespiral.



−0.92 −0.9199−0.9199−0.9199−0.9199−0.9199−0.9198

0.608

0.61

0.612

0.614

0.616

0.618

0.62

0.622

0.624

0.626

0.628

c

γ

Figure 4.13 Enlargement of the most left spiral of the IV-characteristics for S = −0.1456and α = 0.18. Curves from left to right: J = 1, J = 1.00001, J = 1.000015, J = 1.00002 &J = 1.00003.


June 2005


4.4 Unequal Junctions (J < 1)

Although for the situation of J significant smaller than 1 the solution branches are not thor-oughly investigated, there are still some results to show for J slightly smaller than 1. In Figure4.14(a), the IV-characteristic is shown for J just below 1 (J = 0.995). This IV-characteristicconsists of (1) the primary branch, which follows the perturbation relation (2.37) for smallparameter values, (2) the (small) loop in the lower right part, on which the parameter valueslie for the solutions in a [1|1,−1] state and (3) a closed loop with two connected spirals in-side. The first two branches are straightforward related to the equivalent branches for J = 1.The closed loop with the two connected spirals inside, however, needs some more explanation.

For two identical junctions (i.e. J = 1), the IV-characteristics consisted of the first twoaforementioned branches and the headphone shaped curve, as visible in Figure 4.10(a). Thisheadphone shaped curve is enlarged in Figure 4.10(b). An enlargement of the closed loopwith the two connected spirals inside is displayed in Figure 4.14(b). Comparison of two thespirals of the headphone curve for J = 1 with the connected spirals for J = 0.995 learns thatfor both values of J , the left one spirals counterclockwise while the right one spirals clockwise.Hence, the spiraling direction of these spirals is not affected by the decrease of J from 1 to0.995, in contrast to what happens to the spiraling direction of the most left spiral in Figure4.10 if J is increased from 1. What happens here is that, if J decreases from 1, the lowestturning point of the left spiral of the headphone curve (of Figure 4.10(b)) and the most leftturning point of the right spiral of the headphone curve (of Figure 4.10(b)) move towardseach other until they collide. If J is decreased further, the branches at the collision pointsplit up again, where the two parts of the branches above the collision point stay connectedas well as the two parts of the branches below the collision point. The upper connection nowforms the connection between the two spirals and the lower connection forms the closed loop.

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

c

γ

(1) (2)

(3)

(a) overview of the solution branches

−0.9 −0.88 −0.86 −0.84 −0.82

0.55

0.6

0.65

0.7

0.75

(b) enlargement of the closed loop of the IV-characteristic with the spirals inside (the dotted curveis the primary branch)

Figure 4.14 The IV-characteristic for J = 0.995, S = −0.1456 and α = 0.18.


51

Chapter 5

Conclusion and Recommendations

In this thesis, the dynamics of a fluxon in a two-fold stack of long Josephson junctions istreated. Central point of the analytical research were the equations that govern the dynamicsof a two-fold stack of coupled long Josephson junctions given by system (2.8) (page 9). Thedynamics are considered for a stack in which the dissipative effect of quasi-particles flowingalong the surfaces of the superconductors of the junctions is neglected. Hence, the equationsthat govern the dynamics are analyzed without β-term. Moreover, the perturbations of thesystem were considered to be small (which is naturally true for the dissipative effect of quasi-particle tunnelling across the junctions) in order to perform a perturbation analysis.

For a two-fold stack of junctions in the [1|0] state, the analytical expressions of the travellingwave solutions for small perturbations are derived up to order ε. Also the stability of thesetravelling wave solutions has been investigated.

The numerical results are obtained by using the software package AUTO ([8]), which is able tocalculate the parameter combinations for which there exists a solutions corresponding to the[1|0] state in the two-fold stack. The effect of the coupling on the solution branches has beeninvestigated. Also the effect of the ratio J of the critical currents on the solution brancheshas been investigated for the transition from J larger than 1 to J smaller than 1.

5.1 Analysis

The aim of Chapter 2 was to determine analytical expressions of the travelling wave solutions,which represent the phase differences over both junctions in a two-fold stack. An analysishas been performed for the situation that the stack contains one fluxon in the first junctionand none in the second junction, i.e. the stack is in the so-called [1|0] state. By using regularperturbation theory, the analytical solutions for the phase differences in both junctions aredetermined up to O(ε), where ε is assumed to be small. The leading order term ϕA

0 (ξ) ofthe phase difference over the first junction is given by (2.12) (page 11). This leading orderterm is in fact the phase difference over a single junction without perturbation terms whichcontains a fluxon, i.e. a junction with no quasi-particles tunnelling across the junction andflowing along the surfaces of the superconductors and also no bias current applied. Since thesecond junction does not contain a fluxon, the term ϕB

0 (ξ) of the phase difference over thisjunction of corresponding order to ϕA

0 is 0.

The ε order terms of the phase differences over both junctions are given by ϕA1 (ξ) ((2.39),

page 16) and ϕB1 (ξ) ((2.48), page 18). The term of O(ε) of the solution in the first junction is


June 2005

52 5. Conclusion and Recommendations

not influenced by the coupling between the junctions. This is due to the fact that there is nofluxon in the second junction and hence the leading order term of the phase difference overthe second junction is of order ε. The flowing currents in the second junction do not inducea magnetic field strong enough to influence the flowing currents in the first junction suchthat this influence is reflected in the ϕA

1 (ξ). Therefore, the influence of the phase differenceover the second junction on the phase difference over the first junction will only play a rolein the analytical solution for the first junction, at terms of order εn, n ≥ 2. Hence, theexpression for ϕA

0 (ξ) and ϕA1 (ξ) of the phase difference over the first junction are the same

as the analytical expressions of the leading order term and the ε order term of a single longJosephson junction with similar properties (as has been done in, e.g., ([7])). On the otherhand, the O(ε) term of the analytical expression of phase difference over the second junctionis influenced by the leading order term ϕA

0 (ξ) of the phase difference over the first junction(as can be seen immediately from (2.48).

5.2 Stability

In Chapter 3, the stability of the determined travelling wave solutions for the phase differencesover the first and the second junction is investigated. This has been done by means of analyz-ing the essential spectrum and the point spectrum. The boundary of the essential spectrum(which is related to the stability of the equilibria) is shown to reside in the left half-planeof the complex plane. The boundary of the essential spectrum (for small perturbations) isshown in Figure 3.1 (page 25). Since the boundary of the essential spectrum resides in theleft half-plane, this will not give rise to a (linear) instability of the equilibria. Hence, thetravelling wave solutions converging to these equilibria can be stable, but that still dependson the analysis of the point spectrum.

The point spectrum is investigated by using the Evans function formulation. For the unper-turbed system (ε = 0), the point spectrum has been determined which gave the result thatall zeros of the Evans function have zero real part. Hence, the travelling wave solutions ofthe unperturbed system are (linearly) stable. However, the point spectrum of the perturbedsystem (for small perturbations) is more difficult to determine, especially for the situation ofunequal junctions in the stack (i.e. for J 6= 1). It has been shown that for all J , no zerosof the Evans function at the origin move into the right half-plane if the system is perturbed.Furthermore, for the situation of equal junctions (J = 1), it has been shown that no zerosof the Evans function at the end points of the essential spectral can move into the right halfplane by bifurcating out of the essential spectrum if the system is perturbed. However, forthe situation of unequal junctions, the analysis of the Evans function near the end points ofthe essential spectrum did not give conclusive information. It could not be determined if thezeros of the Evans function at the end points of the essential spectrum move into the left orright half plane. Hence, for equal junctions in the two-fold stack, the analytically determinedtravelling wave solutions are linearly stable for sufficiently small perturbations. In order todetermine the stability of these travelling wave solutions for unequal junctions, a differentapproach might be required to perform the analysis of the point spectrum.

5.3 Numerical Results

In Chapter 4, the obtained numerical results are displayed. It has been observed that thederived analytical solutions are in good agreement with the computed numerical results forsmall enough perturbations. Three different situations with respect to the ratio J of the


5.4. Recommendations for Further Research 53

critical currents of the junctions have been investigated.

An interesting phenomenon that occurs if J is larger than 1 is the so-called backbendingeffect. In previous research (e.g. ([16])) this backbending effect has also been observed inthe IV-characteristic. However, in ([16]) the only mentioned solution branch is the closedloop, showing the backbending effect. In this thesis, it has been shown that this single closedloop is not the unique shape of the solution branch for J larger than 1. A branch containinga spiral has also been observed. If the ratio of the critical currents and the coupling werechanged, the solution branches collided and separated again, forming branches of a differentshape. However, the characterization of the solutions of corresponding parts of the branchesbefore and after the collision were similar.

Furthermore, it has been observed that the backbending effect does not always occur for Jlarger than 1, as was believed. For J very close to 1 but still larger than 1, the solutionbranches do not show the backbending effect. If J is increased from 1 to the minimum valuefor which the backbending effect occurs, a rapid change of the solution branches has been seen.

Another interesting phenomenon which occurs for J smaller than 1 is that the velocity of thefluxon is no longer bounded by the lowest Swihart velocity c+, as was the case for J ≥ 1.Nevertheless, the system has a singularity at the lowest Swihart velocity and therefore resultsof this phenomenon have not been obtained. In order to observe this phenomenon, first thesystem should be de-singularized such that the continuation through the singularity at c+

can be done. However, also for J smaller than 1, the solution branches (of solutions travel-ling at a speed less than c+) which corresponded to the branches for J ≥ 1 have been obtained.

For all three situations of J , spirals have been observed in the parameter space. Based onthe characterization of solutions for parameter values close to the centre of each spiral, it isbelieved that the centre of every spiral corresponds to a heteroclinic solution exists.

5.4 Recommendations for Further Research

First point of interest for further research is the inconclusive stability analysis of the travellingwave solutions for unequal junctions. As mentioned in Section 5.2, a different approach mightbe required to proof the stability of the travelling wave solutions for J 6= 1.

Furthermore, since for values of the ratio of the critical currents close to 1 also the numer-ical results have shown a lot of changes in the solution branches, it might be interestingto perform an (analytical) analysis of the system for J close to 1. Also the minimum valueof J close to but larger than 1 for which the backbending effect occurs is something to look for.

Also the existence of the observed spirals in parameter space is interesting. For a single junc-tion with non-zero surface resistance (with non-zero β-term), a spiral also has been observedin parameter space. In ([16]), it has been proven that the existence of that spiral is a resultof the existence of a solution forming a heteroclinic connection between two equilibria for theparameter values at the centre of the spiral, combined with the fact that one of the equilibriais a saddle-focus. Also the existence of the heteroclinic connection itself at the centre of thespiral is proven in ([16]). Therefore, it is interesting to see if for this situation, it can also beproven that the observed spirals are the result of the existence of a heteroclinic solution atthe centre of the spirals. And if so, the existence of the heteroclinic solution itself should be


June 2005

54 5. Conclusion and Recommendations

proven.Furthermore, it might be interesting to see if and how the solution branches change if theratio of the critical currents changes to (significantly) smaller than 1. For that case, solutionsexist which can travel at a speed larger than the lowest Swihart velocity.

Finally, several extensions of the analyzed system can be thought of to investigate. The mostobvious extension is to see what the effect of non-zero surface resistance is on the solution.For that situation, the equations should be analyzed with β-term. Furthermore, differentfluxon states in the two-fold stack could be considered or an extension of the stack itself tomore than two junctions.


55

Appendix A

Travelling Wave Solutions of thePerturbed Sine-Gordon Equation

In this appendix, the travelling wave solutions in the first and the second junction are derived.First, the solution of the unperturbed system is determined. Subsequently the solutions oforder ε in the first and the second junction of the perturbed system are determined.

A.1 Unperturbed System

Consider the equations of a two-fold stack of coupled Josephson junctions. These are givenby

ΦAxx − ΦA

tt − sin(ΦA) = αΦAt − βΦA

xxt + SΦBxx − γ,

ΦBxx − ΦB

tt −sin(ΦB)

J= αΦB

t − βΦBxxt + SΦA

xx − γ,(A.1)

where α, β > 0, 0 ≤ γ ≤ 1 and S ∈ [−1, 0]. Taking β = 0 and scaling α, γ and S by ε,i.e. α = εα, γ = εγ and S = εS, makes it possible to perform a perturbation analysis. LetΦA,B(x, t) ≡ ϕA,B(x − ct) = ϕA,B(ζ). By dropping the tildes on the scaled variables, theperturbed system then turns into

(1− c2)ϕAζζ − sin(ϕA) = ε(−αcϕA

ζ + SϕBζζ − γ),

(1− c2)ϕBζζ −

sin(ϕB)J

= ε(−αcϕBζ + SϕA

ζζ − γ).(A.2)

If ε = 0, the system is unperturbed and the homogeneous equations are

ϕAζζ −

sin(ϕA)1− c2

= 0,

ϕBζζ −

sin(ϕB)J(1− c2)

= 0.(A.3)

When there is no fluxon in the second junction, the solution to the unperturbed system isgiven by ϕB(ζ) = ϕB

0 (ζ) ≡ 0 and ϕA(ζ) = ϕA0 (ζ), where ϕA

0 (ζ) should satisfy

(ϕA0 )ζ = ψA

0 ,

(ψA0 )ζ =

sin(ϕA0 )

1− c2.

(A.4)


June 2005

56 A. Travelling Wave Solutions of the Perturbed Sine-Gordon Equation

Figure A.1 The travelling wave solution ϕA0 (ξ) of the unperturbed system.

The Hamiltonian of these equations is given by H(ϕA0 , ψA

0 ) =(ψA

0 )2

2+

cos(ϕA0 )− 1

1− c2. Setting

H(ϕA0 , ψA

0 ) = 0 and substituting ψA0 = (ϕA

0 )ζ results in

((ϕA0 )ζ)2

2=

1− cos(ϕA0 )

1− c2, so (ϕA

0 )ζ =dϕA

0

dζ=

√2(1− cos(ϕA

0 ))1− c2

. (A.5)

Separating variables gives

∫dζ =

∫ √1− c2

2(1− cos(ϕA0 ))

dϕA0 =

∫ √√√√ 1− c2

2(2 sin2(ϕA02 ))

dϕA0 ,

ζ + ζ0 =√

1− c2

∫1

2 sin(ϕA02 )

dϕA0 =

√1− c2 ln(tan(

ϕA0

4)),

⇒ ϕA0 (ζ) = 4 arctan( e

ζ+ζ0√1−c2 ) = 4 arctan( eξ+ξ0)

where ξ = x−ct√1−c2

, ξ0 = ζ0√1−c2

.

(A.6)

Substituting ϕA,B(ζ) = ϕA,B(ξ(ζ)) in (A.2) leads to

ϕAξξ − sin(ϕA) = ε

(− αc√

1−c2ϕA

ξ + S1−c2

ϕBξξ − γ

),

ϕBξξ − sin(ϕB)

J = ε(− αc√

1−c2ϕB

ξ + S1−c2

ϕAξξ − γ

).

(A.7)

Now, ϕA0 (ξ) = 4 arctan( eξ), ϕB

0 = 0 are the travelling wave solutions to the unperturbedsystem

{(ϕA)ξξ − sin(ϕA) = 0,

(ϕB)ξξ − sin(ϕB)J = 0,

(A.8)

under the condition ϕA(0) = π to get rid of an arbitrary phase shift. This gives ξ0 = 0.Furthermore,

limξ→−∞

ϕA0 (ξ) = 0,

limξ→∞

ϕA0 (ξ) = 2π.

(A.9)

The travelling wave solution to the perturbed system is assumed to be of the form of a series


A.1. Unperturbed System 57

in ε, i.e.{

ϕAε = ϕA

0 (ξ) + εϕA1 (ξ) + O(ε2),

ϕBε = ϕB

0 (ξ) + εϕB1 (ξ) + O(ε2) = εϕB

1 (ξ) + O(ε2).(A.10)

The fixed points of the perturbed system are divided in two classes and given by

ϕA = ϕA2n = arcsin(εγ) + 2nπ,

ϕB = ϕB2n = arcsin(εγJ) + 2nπ,

n = 0,±1,±2, ..., (A.11)

andϕA = ϕA

2n+1 = π − arcsin(εγ) + 2nπ,ϕB = ϕB

2n+1 = π − arcsin(εγJ) + 2nπ,n = 0,±1,±2, ... (A.12)

Since a fluxon corresponds to a solution which connects ϕA,B2n and ϕA,B

2(n+1). By imposingϕA

ε (0) = π + arcsin(εγ), only the fixed points of the former class are relevant. To get rid ofthe arbitrary phase shift as for the homogeneous situation, the solution ϕA,B

ε (ξ) should satisfythe conditions

limξ→−∞

ϕAε (ξ) = arcsin(εγ),

limξ→∞

ϕAε (ξ) = arcsin(εγ) + 2π,

limξ→±∞

ϕBε (ξ) = arcsin(εγJ).

(A.13)

Substitution of the ϕA,Bε series in these conditions results in

limξ→±∞

ϕA1 = γ and lim

ξ→±∞ϕB

1 = γJ. (A.14)

So, defining ϕA1 = ϕA

1 + γ and ϕB1 = ϕB

1 + γJ , by substituting (A.10)1,2 in (A.7)1,2 theequations for ϕA,B

1 (ξ) becomes (by collecting terms of O(ε))

(ϕA1 )ξξ − cos(ϕA

0 )ϕA1 = − αc√

1− c2(ϕA

0 )ξ − γ(1− cos(ϕA0 )),

(ϕB1 )ξξ − ϕB

1

J=

S

1− c2(ϕA

0 )ξξ,(A.15)

under the conditions

limξ→±∞

ϕA1 (ξ) = 0,

limξ→±∞

ϕB1 (ξ) = 0,

(A.16)

and ϕA1 (0) = 0. The equations for ϕA,B

1 (ξ) can be written as two matrix equations

(ΨA1 )ξ(ξ) = AA(ξ)ΨA

1 (ξ) + HA(ξ),(ΨB

1 )ξ(ξ) = ABΨB1 (ξ) + HB(ξ),

(A.17)

where

ΨA,B1 (ξ) =

[ϕA,B

1

(ϕA,B1 )ξ

], AA(ξ) =

[0 1

cos(ϕA0 (ξ)) 0

], AB =

[0 11J

0

],

HA(ξ) =

[0

− αc√1− c2

(ϕA0 )ξ − γ(1− cos(ϕA

0 ))

],

and

HB(ξ) =

0S

1− c2(ϕA

0 )ξξ

.

(A.18)


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A.2 Solution in the First Junction

First, consider (ΨA1 )ξ = AA(ξ)ΨA

1 (ξ) + HA(ξ) with ϕA1 (0) = 0 and lim

ξ→±∞ϕA

1 (ξ) = 0. Let

X(ξ) be a fundamental matrix solution of the homogeneous equation (ΨA1 )ξ = AA(ξ)ΨA

1 (ξ).The columns of X(ξ) consist of linearly independent solutions of the homogeneous equation.Since the homogeneous equation for ΨA

1 (ξ) equals the linearization of the unperturbed systemaround ϕA

0 , the homogeneous equation is solved by the derivative of ϕA0 (ξ). Hence, the first

column is given by (ϕA0 )ξ and (ϕA

0 )ξξ (up to a constant). The second column can be determinedby using the fact that det(X(ξ)) = constant 6= 0, which can easily be seen from the derivativeof the determinant. Let

X(ξ) =[

ϕ1(ξ) ϕ2(ξ)(ϕ1)ξ(ξ) (ϕ2)ξ(ξ)

], (A.19)

where ϕ1,2(ξ) solve the homogeneous equation (ϕA1 )ξξ = cos(ϕA

0 )ϕA1 . The determinant of X(ξ)

is given by

det(X(ξ)) = ϕ1(ξ)(ϕ2)ξ(ξ)− (ϕ1)ξ(ξ)ϕ2(ξ). (A.20)

Taking the derivative with respect to ξ results in

ddξ det(X(ξ)) = (ϕ1)ξ(ξ)(ϕ2)ξ(ξ) + ϕ1(ξ)(ϕ2)ξξ(ξ)− (ϕ1)ξξ(ξ)ϕ1(ξ) +

− (ϕ1)ξ(ξ)(ϕ2)ξ = ϕ1(ξ)(ϕ2)ξξ(ξ)− (ϕ1)ξξ(ξ)ϕ2(ξ).(A.21)

Since ϕ1(ξ) and ϕ2(ξ) solve the homogeneous equation, this leads to

ϕ1(ξ)(ϕ2)ξξ(ξ)− (ϕ1)ξξ(ξ)ϕ2(ξ) = ϕ1(ξ) cos(ϕA0 )ϕ2(ξ)− cos(ϕA

0 )ϕ1(ξ)ϕ2(ξ) = 0. (A.22)

So ddξ det(X(ξ)) = 0, thus det(X(ξ)) = constant, independent of ξ. Taking this constant

equal to 1, in accordance with Liouville’s theorem of phase conservation and substitutingϕ1(ξ) = sech(ξ) and ϕ2(ξ) = sech(ξ)z(ξ), then det(X(ξ)) = sech(ξ)2zξ(ξ) = 1. Solving thisfor z(ξ), ϕ2(ξ) is known and X(ξ) is now given by

X(ξ) =[

sech(ξ) 12(ξ sech(ξ) + sinh(ξ))

( sech(ξ))′ 12(ξ sech(ξ) + sinh(ξ))′

]. (A.23)

Furthermore X(0) = I2, so X(ξ) is also the so-called principal matrix solution at initial“time” ξ0 = 0. Evaluating the elements of X(ξ) at ±∞ shows that

limξ→∞

X(ξ) =[

0 ∞0 ∞

]and lim

ξ→−∞X(ξ) =

[0 −∞0 ∞

]. (A.24)

If now[

p1

p2

]is an initial state vector at ξ = 0, then X(ξ)

[p1

p2

]is bounded on R+ if p2 = 0.

So, the stable subspace Ssub = {p ∈ R2 | supt≥0

|X(t)p| < ∞} consists of multiplications of the

vector [1, 0]T . Let P s(0) be the projection on the stable subspace at ξ = 0, then

P s(0) =[

1 00 0

](A.25)

and the projection P u(0) ≡ I2 − P s(0) is given by

P u(0) =[

0 00 1

]. (A.26)


A.2. Solution in the First Junction 59

Figure A.2 Plot of the elements of the fundamental matrix solution X(ξ). Solid curve:sech(ξ), dotted curve: d

dξ sech(ξ), dashed curve: 12( sech(ξ) + sinh(ξ)), dashed-dotted curve:

ddξ

12(ξ sech(ξ) + sinh(ξ)).

Likewise on R−, X(ξ)[

p1

p2

]is bounded on R− if p2 = 0. So, the unstable subspace Usub =

{p ∈ R2 | supt≤0

|X(t)p| < ∞} consists of multiplications of the vector [1, 0]T and therefore the

projection P u(0) on the unstable subspace at ξ = 0 is

P u(0) =[

1 00 0

](A.27)

with the projection P s(0) ≡ I2 − P u(0),

P s(0) =[

0 00 1

]. (A.28)

Exponential Dichotomies

The equation (ΨA1 )ξ = AA(ξ)ΨA

1 now possesses an exponential dichotomy on R+ with theprojection P s(0) and an exponential dichotomy on R− with the projection P u(0). Thismeans that there exists an α, β, K,L, α, β, K and L such that

{ |X(t)P s(0)X−1(s)| ≤ K e−α(t−s), t ≥ s ≥ 0, K, α > 0,

|X(t)P u(0)X−1(s)| ≤ L e−β(s−t), s ≥ t ≥ 0, L, β > 0(A.29)


June 2005


and{|X(t)P u(0)X−1(s)| ≤ K eα(t−s), t ≤ s ≤ 0, K, α > 0,

|X(t)P s(0)X−1(s)| ≤ L eβ(s−t), s ≤ t ≤ 0, L, β > 0(A.30)

holds. To see that the inequalities hold, take X(ξ) =[

f(ξ) g(ξ)f ′(ξ) g′(ξ)

], where f(ξ) = sech(ξ),

g(ξ) = 12(ξ sech(ξ) + sinh(ξ)) and the prime denotes differentiation with respect to ξ. Inves-

tigating the matrix elements of X(ξ) (for ξ ≥ 0) leads to

|f(ξ)| = sech(ξ) =2

eξ + e−ξ≤ 2

eξ= 2 e−ξ,

|f ′(ξ)| = sech(ξ) tanh(ξ) =2( eξ − e−ξ)( eξ + e−ξ)2

≤ 2( eξ + e−ξ)( eξ + e−ξ)2

=2

eξ + e−ξ=

= sech(ξ) ≤ 2 e−ξ,

|g(ξ)| = 12ξ sech(ξ) +

12

sinh(ξ) =ξ

eξ + e−ξ+

14( eξ − e−ξ)

≤ ξ

eξ + e−ξ+

14

eξ ≤ 1 +14

eξ ≤ 114

eξ,

|g′(ξ)| = 12

sech(ξ)− 12ξ sech(ξ) tanh(ξ) +

12

cosh(ξ) =

=ξ

eξ + e−ξ− ξ( eξ − e−ξ)

( eξ + e−ξ)2+

14( eξ + e−ξ)

≤ ξ

eξ + e−ξ− ξ( eξ + e−ξ)

( eξ + e−ξ)2+

14( eξ + e−ξ) =

=2ξ

eξ + e−ξ+

14( eξ + e−ξ) ≤ 2 +

12

eξ ≤ 212

eξ.

(A.31)

Evaluating X(t)P s(0)X−1(s) gives

X(t)P s(0)X−1(s) =[

f(t)g′(s) −f(t)g(s)f ′(t)g′(s) −f ′(t)g(s)

], (A.32)

where t ≥ s ≥ 0 is assumed. Since |f ′(ξ)| ≤ |f(ξ)|, it follows that |X(t)P s(0)X−1(s)| =|f(t)||g′(s)|+ |f(t)||g(s)| and thus

|X(t)P s(0)X−1(s)| ≤ 2 e−t(114

es + 212

es) = 712e−t es = 7

12

e−(t−s). (A.33)

Evaluating X(t)P u(0)X−1(s) gives

X(t)P u(0)X−1(s) =[ −g(t)f ′(s) g(t)f(s)−g′(t)f ′(s) g′(t)f(s)

], (A.34)

where s ≥ t ≥ 0 is assumed. Here, |g(t)|, |g′(t)| ≤ 212 et, so

|X(t)P u(0)X−1(s)| ≤ 212

et(|f(s)|+ |f ′(s)|) ≤ 212

et4 e−s = 9 et e−s = 9 e−(s−t). (A.35)

Now, the inequalities of the exponential dichotomy on R+ hold with α = β = 1, K = 712

and L = 9. Similarly, the same analysis on R− can be performed, which shows that theinequalities of the exponential dichotomy on R− hold with α = β = 1, K = 71

2 and L = 9.




The general solution of (ΨA1 )ξ = AA(ξ)ΨA

1 (ξ) + HA(ξ) is given by the variation of constantsformula, i.e.

ΨA1 (ξ) = X(ξ)X−1(σ)ΨA

1 (σ) + X(ξ)∫ ξ

σX−1(τ)HA(τ) dτ. (A.36)

Furthermore, the projections P s,u(ξ) = X(ξ)P s,u(0)X−1(ξ) and P u,s(ξ) = X(ξ)P u,s(0)X−1(ξ)are defined, where P u(ξ) is the projection on the stable subspace for ξ ≥ 0 and P u(ξ) theprojection on the unstable subspace for ξ ≤ 0. Let ΨA

1+(ξ) be a solution on R+. From the

definitions of the projections P s,u(ξ) it follows that

ΨA1

+(ξ) = P s(ξ)ΨA

1+(ξ) + P u(ξ)ΨA

1+(ξ) =

= X(ξ)P s(0)X−1(ξ)ΨA1

+(ξ) + X(ξ)P u(0)X−1(ξ)ΨA

1+(ξ). (A.37)

Then (by setting σ = 0 in (A.36))

ΨA1

+(ξ) = X(ξ)P s(0)X−1(0)ΨA

1+(0) + X(ξ)P s(0)

∫ ξ

0X−1(τ)HA(τ) dτ +

+ X(ξ)P u(0)X−1(0)ΨA1

+(0) + X(ξ)P u(0)

∫ ξ

0X−1(τ)HA(τ) dτ. (A.38)

Since P s(0) is the projection on the stable manifold at ξ = 0 it follows that ΨA1

+(ξ) is boundedon R+ if the unstable contribution to ΨA

1+(ξ) is zero, i.e.

P u(0)X−1(0)ΨA1

+(0) +

∫ ∞

0P u(0)X−1(τ)HA(τ) dτ = 0

⇒ P u(0)ΨA1

+(0) = −

∫ ∞

0P u(0)X−1(τ)HA(τ). dτ

(A.39)

Working out this condition results in

(ϕA1

+)ξ(0) =

4αc + γπ√

1− c2

2√

1− c2. (A.40)

Hence, provided condition (A.39) is satisfied, ΨA1

+(ξ) is a bounded solution on R+. Then,since the value of σ is free to choose,

P s(ξ)ΨA1

+(ξ) = X(ξ)P s(0)

[X−1(σ)ΨA

1+(σ) +

∫ ξ


]

(σ = 0)

=X(ξ)P s(0)

[ΨA

1+(0) +

∫ ξ

0X−1(τ)HA(τ) dτ

] (A.41)

(note that X−1(0) = I2) and

P u(ξ)ΨA1

+(ξ) = X(ξ)P u(0)

[X−1(σ)ΨA

1+(σ) +

∫ ξ


]

(σ = ∞)

=X(ξ)P u(0)

∫ ξ

∞X−1(τ)HA(τ) dτ,

(A.42)

where the first term vanishes if σ goes to infinity because of the exponential dichotomy onR+. Combining the terms P s(ξ)ΨA

1+(ξ) and P u(ξ)ΨA

1+(ξ) gives the bounded solution on R+,

ΨA1

+(ξ) = X(ξ)

[P s(0)ΨA

1+(0) +

∫ ξ

0P s(0)X−1(τ)HA(τ) dτ +

−∫ ∞


], ξ ≥ 0.

(A.43)


June 2005



The same analysis, as for ξ ≥ 0, can be performed on R−. Let ΨA1−(ξ) be a solution on R−.

Then ΨA1−(ξ) can be split up by using the projections P u,s(ξ),

ΨA1−(ξ) = P s(ξ)ΨA

1−(ξ) + P u(ξ)ΨA

1−(ξ) =

= X(ξ)P s(0)X−1(ξ)ΨA1−(ξ) + X(ξ)P u(0)X−1(ξ)ΨA

1−(ξ). (A.44)

Then (by setting σ = 0 in (A.36))

ΨA1−(ξ) = X(ξ)P s(0)X−1(0)ΨA

1−(0) + X(ξ)P s(0)

∫ ξ

0X−1(τ)HA(τ) dτ +

+ X(ξ)P u(0)X−1(0)ΨA1−(0) + X(ξ)P u(0)

∫ ξ

0X−1(τ)HA(τ) dτ. (A.45)

Since P u(0) is the projection on the unstable manifold at ξ = 0, ΨA1−(ξ) is bounded on R− if

P s(0)X−1(0)ΨA1−(0) +

∫ −∞

0P s(0)X−1(τ)HA(τ) dτ = 0

⇒ P s(0)ΨA1−(0) =

∫ 0

−∞P s(0)X−1(τ)HA(τ) dτ.

(A.46)

Working out this conditions gives

(ϕA1−)ξ(0) =

−4αc− γπ√

1− c2

2√

1− c2. (A.47)

Provided condition (A.46) is satisfied, ΨA1−(ξ) is a bounded solution on R−. Then, by letting

σ go to −∞ in the first term of the split up expression of ΨA1−(ξ),

P s(ξ)ΨA1−(ξ) = X(ξ)P s(0)

[X−1(σ)ΨA

1−(σ) +

∫ ξ


]=

(σ = −∞)

=X(ξ)P s(0)

∫ ξ

−∞X−1(τ)HA(τ) dτ

(A.48)

where the first term vanishes if σ goes to −∞, because of the exponential dichotomy on R−.Choosing σ = 0 for the second term gives

P u(ξ)ΨA1−(ξ) = X(ξ)P u(0)

[X−1(σ)ΨA

1−(σ) +

∫ ξ


]=

(σ = 0)

=X(ξ)P u(0)

[ΨA

1−(0) +

∫ ξ

0X−1(τ)HA(τ) dτ

].

(A.49)

Again, combining the expression of P s(ξ)ΨA1−(ξ) and P u(ξ)ΨA

1−(ξ) gives the bounded solution

on R−,

ΨA1−(ξ) = X(ξ)

[∫ ξ

−∞P s(0)X−1(τ)HA(τ) dτ + P u(0)ΨA

1−(0) +

−∫ 0


], ξ ≤ 0.

(A.50)



Bounded Solution for ξ ∈ RNow, the bounded solution on R can be constructed from the two bounded solutions on R+

and R− as follows

ΨA1 (ξ) =

{ΨA

1−(ξ), ξ < 0,

ΨA1

+(ξ), ξ ≥ 0.(A.51)

The necessary condition for the bounded solution to exist on R is that the bounded solutionson R+ and R− match at ξ = 0, i.e. if ΨA

1+(0) = ΨA

1−(0)(= ΨA

1 (0)). Setting the two solutionsequal to each other at ξ = 0 results in the requirement

P s(0)ΨA1

+(0)−

∫ ∞

0P u(0)X−1(τ)HA(τ) dτ =

∫ 0

−∞P s(0)X−1(τ)HA(τ) dτ + P u(0)ΨA

1−(0).

(A.52)

Since P s(0) = P u(0), this requirement is equivalent to the requirements P s(0)ΨA1

+(0) =P s(0)ΨA

1−(ξ) and

∫ ∞

−∞P u(0)X−1(τ)HA(τ) dτ = 0. (A.53)

Working out this integral condition results in

−4αc + γπ√

1− c2

√1− c2

= 0 ⇒ c(γ) = − γπ√16α2 + γ2π2

, (A.54)

where the former requirement gives ϕA1

+(0) = ϕA1−(0) (= ϕA

1 (0)) which is imposed to be 0.Furthermore, with c(γ) of the required form, (ϕA

1 )ξ(0) (= ϕA1

+(0) = ϕA1−(0)) becomes 0. Now

with ΨA1 (0) = [0, 0]T , ΨA

1 (ξ) is given by

ΨA1 (ξ) =

X(ξ)[∫ ξ

−∞P s(0)X−1(τ)HA(τ) dτ −

∫ 0


], ξ < 0,

X(ξ)[∫ ξ

0P s(0)X−1(τ)HA(τ) dτ −

∫ ∞


], ξ ≥ 0,

(A.55)

where HA(ξ) = HA(ξ)|c=c(γ) =

0

γ

(12π sech(ξ) + cos(4 arctan( eξ))− 1

).

For ξ ≥ 0, the explicit form of ϕA1 (ξ) is given by

ϕA1 (ξ) = sech(ξ)

∫ ξ

0X−1

12 (τ)HA21(τ) dτ +

− 12(ξ sech(ξ) + sinh(ξ))

∫ ∞

ξX−1

22 (τ)HA21(τ) dτ,

(A.56)

where working out the integral of the first term of (A.56) leads to∫ ξ

0X−1

12 (τ)HA21(τ) dτ =

= γ

∫ ξ

0

4 sinh(τ) e2τ

( e2τ + 1)2− π

4(sinh(τ) sech(τ) + τ sech(τ)2) +

4τ sech(τ) e2τ

( e2τ + 1)2dτ =

= γ

[( eξ − 1)2

e2ξ + 1− πξ( e2ξ − 1)

4( e2ξ + 1)+

∫ ξ

0

8τ e3τ

( e2τ + 1)3dτ

],

(A.57)


June 2005


and working out the integral of the second term of (A.56) gives∫ ∞

ξX−1

22 (τ)HA21(τ) dτ = γ

∫ ∞

ξ

12π sech(τ)2 − 8 sech(τ) e2τ

( e2τ + 1)2dτ =

= γ

[ −π e2ξ

e2ξ + 1− 2( eξ − e3ξ)

( e2ξ + 1)2+ 2 arctan( eξ)

].

(A.58)

So, the complete expression of ϕA1 (ξ) for ξ ≥ 0 is given by

ϕA1 (ξ) = γ sech(ξ)

[( eξ − 1)2

e2ξ + 1− πξ( e2ξ − 1)

4( e2ξ + 1)+

∫ ξ

0

8τ e3τ

( e2τ + 1)3dτ

]+

+12γ(ξ sech(ξ) + sinh(ξ))

[π e2ξ

e2ξ + 1+

2( eξ − e3ξ)( e2ξ + 1)2

− 2 arctan( eξ)]

, ξ ≥ 0.

(A.59)

For ξ < 0, ϕA1 (ξ) is given by

ϕA1 (ξ) =

12(ξ sech(ξ) + sinh(ξ))

∫ ξ

−∞X−1

22 (τ)HA21(τ) dτ +

− sech(ξ)∫ 0

ξX−1

12 (τ)HA21(τ) dτ,

(A.60)

where the integral of the first term of (A.60) is∫ ξ

−∞X−1

22 (τ)HA21(τ) dτ = γ

∫ ξ

−∞

12π sech(τ)2 − 8 sech(τ) e2τ

( e2τ + 1)2dτ =

= γ

[π e2ξ

e2ξ + 1+

2( eξ − e3ξ)( e2ξ + 1)2

− 2 arctan( eξ)]

,

(A.61)

and the second term of (A.60) is equivalent to the first term of (A.56). The complete expres-sion of ϕA

1 (ξ) for ξ < 0 is then given by

ϕA1 (ξ) =

12γ(ξ sech(ξ) + sinh(ξ))

[π e2ξ

e2ξ + 1+

2( eξ − e3ξ)( e2ξ + 1)2

− 2 arctan( eξ)]

+

+ γ sech(ξ)[( eξ − 1)2

e2ξ + 1− πξ( e2ξ − 1)

4( e2ξ + 1)+

∫ ξ

0

8τ e3τ

( e2τ + 1)3dτ

].

(A.62)

This shows that the expressions of ϕA1 (ξ) for ξ ≥ 0 (A.59) and ξ < 0 (A.62) are the same and

the expression of ϕA1 (ξ), as in (A.59), can be used for ξ ∈ R.



Figure A.3 ϕA1 (ξ) for γ = −2 (solid curve), γ = −1 (dotted curve), γ = 1 (dashed curve)

and γ = 2 (dashed-dotted curve).

Figure A.4 (ϕA1 )ξ(ξ) for γ = −2 (solid curve) and γ = 1 (dotted curve).


June 2005


A.3 Solution in the Second Junction

Next, consider (ΨB1 )ξ = ABΨB

1 (ξ) + HB(ξ) with limξ→±∞ ϕB1 (ξ) = 0 where

AB =

[0 11J

0

]and HB(ξ) =

0S

1− c2(ϕA

0 (ξ))ξξ

. (A.63)

A fundamental matrix solution X(ξ) of the homogeneous equation (ΨB1 )ξ = ABΨB

1 (ξ) is givenby

X(ξ) =

e

ξ√J e−

ξ√J

eξ√J√J

− e− ξ√

J√J

, with X−1(ξ) =

12

[e−

ξ√J

√J e−

ξ√J

eξ√J

√J e

ξ√J

], (A.64)

where ±ξ/√

J are the eigenvalues of AB. Define Y (ξ) = X(ξ)X−1(0). Then Y (ξ) is also afundamental matrix solution of the homogeneous equation since it satisfies Yξ = ABY (ξ),

Yξ(ξ) =d

dξX(ξ)X−1(0) = Xξ(ξ)X−1(0) = ABX(ξ)X−1(0) = ABY (ξ). (A.65)

Furthermore, since Y (0) = X(0)X−1(0) = I2, Y (ξ) is the principal matrix solution at initial“time” ξ0 = 0. This principal matrix solution is given by

Y (ξ) = X(ξ)X−1(0) =

12( e

ξ√J + e−

ξ√J )

12

√J( e

ξ√J − e−

ξ√J )

12√

J( e

ξ√J − e−

ξ√J )

12( e

ξ√J + e−

ξ√J )

, (A.66)

which is in fact the exponential of matrix AB, i.e. Y (ξ) = eABξ. The stable and unstable

directions are given by[1,− 1√

J

]T

and[1,

1√J

]T

, respectively. The projection P s(0) on the

stable direction along the unstable direction at ξ = 0 then has to satisfy

P s(0)

1

− 1√J

=

1

− 1√J

and P s(0)

11√J

=

[00

]. (A.67)

From this, the projection P s(0) can be determined and is given by

P s(0) =

12

−√

J

2− 1

2√

J

12

and P u(0) ≡ I2 − P s(0) =

12

√J

21

2√

J

12

. (A.68)

If now[

p1

p2

]is an initial condition at ξ = 0 with p1, p2 ∈ R, then Y (ξ)P s(0)

[p1

p2

]is

bounded for ξ → ∞ and Y (ξ)P u(0)[

p1

p2

]is bounded for ξ → −∞. So, the projections

P s,u(0) (where P u(0) is the projection on the unstable subspace at ξ = 0 as introduced in theanalysis of the first junction) are equal to P s,u(0), i.e. P s(0) = P s(0) and P u(0) = P u(0).

Now, with the projection P s(0), the equation (ΨB1 )ξ(ξ) = ABΨB

1 (ξ) possesses an exponentialdichotomy on R, i.e. there exists an α, β, K and L such that

{ |Y (t)P s(0)Y −1(s)| ≤ K e−α(t−s), t ≥ s, t, s ∈ R, K, α > 0,

|Y (t)P u(0)Y −1(s)| ≤ L e−β(s−t), s ≥ t, s, t ∈ R, L, β > 0,(A.69)


A.3. Solution in the Second Junction 67

holds. The inequalities follow directly from evaluating the matrix multiplication

Y (t)P s(0)Y −1(s) =

12

e−1√J

(t−s) −12

√J e−

1√J

(t−s)

− 12√

Je−

1√J

(t−s) 12

e−1√J

(t−s)

(A.70)

and

Y (t)P u(0)Y −1(s) =

12

e1√J

(t−s) 12

√J e

1√J

(t−s)

12√

Je

1√J

(t−s) 12

e1√J

(t−s)

. (A.71)

So, the inequalities are satisfied if α = β = 1/√

J and K = L = max(

12 +

√J

2 , 12√

J+ 1

2

).


The general solution of (ΨB1 )ξ = AB(ξ)ΨB

1 (ξ) + HB(ξ) is given by

ΨB1 (ξ) = Y (ξ)Y −1(σ)ΨB

1 (σ) + Y (ξ)∫ ξ

σY −1(τ)HB(τ) dτ. (A.72)

Let ΨB1

+(ξ) be a solution on R+. By defining the projections P s,u(ξ) on the stable andunstable subspace for ξ ∈ R as P s,u(ξ) = Y (ξ)P s,u(0)Y −1(ξ), the solution ΨB

1+(ξ) can be

split up as

ΨB1

+(ξ) = P s(ξ)ΨB

1+(ξ) + P u(ξ)ΨB

1+(ξ) =

= Y (ξ)P s(0)Y −1(ξ)ΨB1

+(ξ) + Y (ξ)P u(0)Y −1ΨB

1+(ξ). (A.73)

Then by setting σ = 0 in (A.72),

ΨB1

+(ξ) = Y (ξ)P s(0)Y −1(0)ΨB

1+(0) + Y (ξ)P s(0)

∫ ξ

0Y −1(τ)HB(τ) dτ +

+ Y (ξ)P u(0)Y −1(0)ΨB1

+(0) + Y (ξ)P u(0)

∫ ξ

0Y −1(τ)HB(τ) dτ. (A.74)

Since P s(0) is the projection on the stable manifold at ξ = 0, the solution ΨB1

+(ξ) is boundedon R+ if the unstable contribution to the solution is zero, i.e. if the following holds

P u(0)Y −1(0)ΨB1

+(0) +

∫ ∞

0P u(0)Y −1(τ)HB(τ) dτ = 0, (A.75)

so P u(0)ΨB1

+(0) = −

∫ ∞

0P u(0)Y −1(τ)HB(τ) dτ should hold. Working out this integral

condition results in

12ϕB

1+(0) +

12

√J(ϕB

1+)ξ(0) =

∫ ∞

0−1

2

√J e−

τ√J

S

1− c2(ϕA

0 (τ))ττ dτ,

12√

JϕB

1+(0) +

12(ϕB

1+)ξ(0) =

∫ ∞

0−1

2e−

τ√J

S

1− c2(ϕA

0 (τ))ττ dτ.

(A.76)


June 2005


Provided condition (A.75) is satisfied, ΨB1

+(ξ) is a bounded solution on R+. Then, since thevalue of σ is free to choose,

P s(ξ)ΨB1

+(ξ) = Y (ξ)P s(0)

[Y −1(σ)ΨB

1+(σ) +

∫ ξ

σY −1(τ)HB(τ) dτ

]

(σ = 0)

=Y (ξ)P s(0)

[ΨB

1+(0) +

∫ ξ

0Y −1(τ)HB(τ) dτ

] (A.77)

and

P u(ξ)ΨB1

+(ξ) = Y (ξ)P u(0)

[Y −1(σ)ΨB

1+(σ) +

∫ ξ


]

(σ = ∞)

=Y (ξ)P u(0)

∫ ξ

∞Y −1(τ)HB(τ) dτ,

(A.78)

where the first terms goes to 0 if σ goes to∞ because of the exponential dichotomy. CombiningP s(ξ)ΨB

1+(ξ) and P u(ξ)ΨB

1+(ξ) gives the complete expression of the bounded solution on R+,

ΨB1

+(ξ) = Y (ξ)

[P s(0)ΨB

1+(0) +

∫ ξ

0P s(0)Y −1(τ)HB(τ) dτ +

−∫ ∞


], ξ ≥ 0. (A.79)


Likewise on R−, let ΨB1−(ξ) be a solution on R− which can be split up with the projections

P s,u(ξ),

ΨB1−(ξ) = P s(ξ)ΨB

1−(ξ) + P u(ξ)ΨB

1−(ξ). (A.80)

Then by setting σ = 0 in (A.72),

ΨB1−(ξ) = Y (ξ)P s(0)Y −1(0)ΨB

1−(0) + Y (ξ)P s(0)

∫ ξ

0Y −1(τ)HB(τ) dτ +

+ Y (ξ)P u(0)Y −1(0)ΨB1−(0) + Y (ξ)P u(0)

∫ ξ

0Y −1(τ)HB(τ) dτ. (A.81)

Then, ΨB1−(ξ) is bounded on R− if the unstable contribution to the solution is zero if ξ goes

to −∞, i.e. if

P s(0)Y −1(0)ΨB1−(0) +

∫ −∞

0P s(0)Y −1(τ)HB(τ) dτ = 0, (A.82)

so P s(0)ΨB1−(0) =

∫ 0

−∞P s(0)Y −1(τ)HB(τ) dτ should hold. Working out this integral condi-

tion results in

12ϕB

1−(0)− 1

2

√J(ϕB

1−)ξ(0) =

∫ 0

−∞−1

2

√J e

τ√J

S

1− c2(ϕA

0 (τ))ττ dτ,

− 12√

JϕB

1−(0) +

12(ϕB

1−)ξ(0) =

∫ 0

−∞

12

eτ√J

S

1− c2(ϕA

0 (τ))ττ dτ.

(A.83)



Provided condition (A.82) is satisfied, ΨB1−(ξ) is a bounded solution on R−. By letting σ go

to −∞, it follows that

P s(ξ)ΨB1−(ξ) = Y (ξ)P s(0)

[Y −1(σ)ΨB

1−(σ) +

∫ ξ


]

(σ = −∞)

=Y (ξ)P s(0)

∫ ξ

−∞Y −1(τ)HB(τ) dτ,

(A.84)

where the first term vanishes because of the exponential dichotomy. Choosing σ = 0 forP u(ξ)ΨB

1−(ξ) leads to

P u(ξ)ΨB1−(ξ) = Y (ξ)P u(0)

[Y −1(σ)ΨB

1−(σ) +

∫ ξ


]

(σ = 0)

=Y (ξ)P u(0)

[ΨB

1−(0) +

∫ ξ

0Y −1(τ)HB(τ) dτ

].

(A.85)

So, the complete expression of ΨB1−(ξ) is given by

ΨB1−(ξ) = Y (ξ)

[∫ ξ

−∞P s(0)Y −1(τ)HB(τ) dτ + P u(0)ΨB

1−(0) +

−∫ 0


], ξ ≤ 0. (A.86)

Bounded Solution for ξ ∈ RThe bounded solution on R can be constructed from the two bounded solutions on R+ andR− as

ΨB1 (ξ) =

{ΨB

1−(ξ), ξ < 0,

ΨB1

+(ξ), ξ ≥ 0.(A.87)

The necessary condition for a bounded solution to exist on R is that the bounded solutionson R+ and R− match at ξ = 0, i.e. if ΨB

1+(0) = ΨB

1−(0) = ΨB

1 (0). Setting the boundedsolutions on R+ and on R− equal to each other at ξ = 0 gives

P u(0)ΨB1−(0) +

∫ 0

−∞P s(0)Y −1(τ)HB(τ) dτ =

= P s(0)ΨB1

+(0)−

∫ ∞

0P u(0)Y −1(τ)HB(τ) dτ,

(A.88)

where now ΨB1−(0) = ΨB

1+(0) = ΨB

1 (0) = [ϕB1 (0), (ϕB

1 )ξ(0)]T . Evaluating this matchingrequirement results in

√

J(ϕB1 )ξ(0)

1√J

ϕB1 (0)

=

−√

J

∫ ∞

0I−B (τ) dτ +

√J

∫ 0

−∞I+B (τ) dτ

−∫ ∞

0I−B (τ) dτ −

∫ 0

−∞I+B (τ) dτ

, (A.89)

where

I±B (τ) =12

e±τ√J

S

1− c2(ϕA

0 (τ))ττ , (A.90)

for which it holds that I±B (−τ) = −I∓B (τ) since (ϕA0 (τ))ττ is odd. Using this relation, it

follows that1√J

ϕB1 (0) = −

∫ ∞

0I−B (τ) dτ −

∫ ∞

0I+B (−τ) dτ = 0 (A.91)


June 2005


and√

J(ϕB1 )ξ(0) = −

√J

∫ ∞

0I−B (τ) dτ +

√J

∫ ∞

0I+B (−τ) dτ = −2

√J

∫ ∞

0I−B (τ) dτ. (A.92)

These initial conditions on ϕB1 (0) and (ϕB

1 )ξ(0) satisfy the conditions (A.75) and (A.82). Thesolutions ΨB

1+(ξ) and ΨB

1−(ξ) are bounded solution on R+ and R− with these initial values.

Now, the bounded solution on R can be constructed using

[ϕB

1 (0)(ϕB

1 )ξ(0)

]=

0

−∫ ∞

0e−

τ√J

S

1− c2(ϕA

0 (τ))ττ dτ

(A.93)

which is then given by

ΨB1 (ξ) =

Y (ξ)

−√

J

(∫ ∞

0I−B (τ) dτ +

∫ ξ

−∞I+B (τ) dτ +

∫ 0

ξI−B (τ) dτ

)

−∫ ∞

0I−B (τ) dτ +

∫ ξ

−∞I+B (τ) dτ −

∫ 0

ξI−B (τ) dτ

, ξ < 0,

Y (ξ)

√J

(∫ ∞

0I−B (τ) dτ −

∫ ξ

0I+B (τ) dτ −

∫ ∞

ξI−B (τ) dτ

)

−∫ ∞

0I−B (τ) dτ +

∫ ξ

0I+B (τ) dτ −

∫ ∞

ξI−B (τ) dτ

, ξ ≥ 0,

(A.94)

=

Y (ξ)

−√

J

∫ 0

ξI−B (τ)− I+

B (τ) dτ

2∫ ξ

−∞I+B (τ) dτ +

∫ 0

ξI+B (τ)− I−B (τ) dτ

, ξ < 0,

Y (ξ)

√J

∫ ξ

0I−B (τ)− I+

B (τ) dτ∫ ξ

0I+B (τ)− I−B (τ) dτ − 2

∫ ∞

ξI−B (τ) dτ

, ξ ≥ 0.

(A.95)

This leads to

ϕB1 (ξ) =

√J e

ξ√J

∫ 0

ξsinh(

τ√J

)S

1− c(γ)2(ϕA

0 (τ))ττ dτ +

+√

J sinh(ξ√J

)∫ ξ

−∞e

τ√J

S

1− c(γ)2(ϕA

0 (τ))ττ dτ, ξ < 0,

−√

J e−ξ√J

∫ ξ

0sinh(

τ√J

)S

1− c(γ)2(ϕA

0 (τ))ττ dτ +

−√

J sinh(ξ√J

)∫ ∞

ξe−

τ√J

S

1− c(γ)2(ϕA

0 (τ))ττ dτ, ξ ≥ 0,

(A.96)

where c is given by c(γ) (A.54).



Figure A.5 ϕB1 (ξ) for J = 1, solid curve: γ = 1, α = 1, S = −2, dotted curve: γ = 2, α =

1, S = −2, dashed curve: γ = 1, α = 2, S = −2 and dashed-dotted curve: γ = 1, α = 1, S =−3.


June 2005


Figure A.6 (ϕB1 )ξ(ξ) for J = 1, solid curve: γ = 1, α = 1, S = −2, dotted curve: γ =

2, α = 1, S = −2, dashed curve: γ = 1, α = 2, S = −2 and dashed-dotted curve: γ = 1, α =1, S = −3.


73

Appendix B

The Spectral Problem

In this appendix the stability of the travelling wave solutions is discussed. First the essentialspectrum is treated in order to determine the stability of the fixed points. Subsequently, thepoint spectrum is used to see if the solutions of the perturbed system are stable.

B.1 The Essential Spectrum

The equations for a stack of two coupled junctions (with β = 0) are given by

ΦAxx − ΦA

tt − sin(ΦA) = ε(αΦAt + SΦB

xx − γ)

ΦBxx − ΦB

tt −sin(ΦB)

J= ε(αΦB

t + SΦAxx − γ).

(B.1)

Writing the equations in a moving coordinate frame by making the transformation ΦA,B(x, t) =ΦA,B(ξ, t) with ξ = x−ct√

1−c2gives

ΦAξξ +

2c√1− c2

ΦAξt − ΦA

tt − sin(ΦA) =

= ε

(− αc√

1− c2ΦA

ξ + αΦAt +

S

1− c2ΦB

ξξ − γ

),

ΦBξξ +

2c√1− c2

ΦBξt − ΦB

tt −sin(ΦB)

J=

= ε

(− αc√

1− c2ΦB

ξ + αΦBt +

S

1− c2ΦA

ξξ − γ

).

(B.2)

Linearizing these equations around the travelling wave solutions ϕA,Bε , i.e. assuming ΦA,B(ξ, t) =

ϕA,Bε (ξ) + uA,B(ξ, t) and obtaining the linearized equations for uA,B(ξ, t), results in

uAξξ +

2c√1− c2

uAξt − uA

tt − cos(ϕAε )uA =

= ε

(− αc√

1− c2uA

ξ + αuAt +

S

1− c2uB

ξξ

)

uBξξ +

2c√1− c2

uBξt − uB

tt −cos(ϕB

ε )J

uB =

= ε

(− αc√

1− c2uB

ξ + αuBt +

S

1− c2uA

ξξ

)

(B.3)

Next, with use of the spectral Ansatz uA,B(ξ, t) = eλtvA,B(ξ), the system turns into{

vAξξ = ω(ξ, λ)vA + η(λ)vA

ξ + νvBξξ,

vBξξ = χ(ξ, λ)vB + η(λ)vB

ξ + νvAξξ,

(B.4)


June 2005

74 B. The Spectral Problem

where

ω(ξ, λ) = λ2 + cos(ϕAε (ξ)) + εαλ,

χ(ξ, λ) = λ2 +cos(ϕB

ε (ξ))J

+ εαλ,

η(λ) =−2cλ− εαc√

1− c2,

ν = εS

1− c2.

(B.5)

This can be rewritten as a first order differential matrix equation of the form ∂∂ξV (ξ) =

A(ξ, λ)V (ξ). Writing equations (B.4) in matrix from gives

1 0 0 00 1 0 −ν0 0 1 00 −ν 0 1

︸︷︷︸B

vA

vAξ

vB

vBξ

ξ

=

0 1 0 0ω(ξ, λ) η(λ) 0 0

0 0 0 10 0 χ(ξ, λ) η(λ)

︸︷︷︸C(ξ,λ)

vA

vAξ

vB

vBξ

⇒

vA

vAξ

vB

vBξ

ξ

= B−1C(ξ, λ)

vA

vAξ

vB

vBξ

= Aε(ξ, λ)

vA

vAξ

vB

vBξ

. (B.6)

So, ∂∂ξV (ξ) = Aε(ξ, λ)V (ξ), where V (ξ) = [vA(ξ), vA

ξ (ξ), vB(ξ), vBξ (ξ)]T . The stability of

the travelling wave solutions is now determined by the operator belonging to the eigenvalueproblem Vξ = Aε(ξ, λ)V . Define T (λ) : U → (d/dξ − Aε(ξ, λ))U . Then the spectrum ofT (λ) consists of those λ for which T (λ) is not invertible, i.e. those λ for which there existsa (non trivial) solution of Vξ(ξ) = Aε(ξ, λ)V (ξ). Let vA,B = CA,B eiκξ, then the essentialspectrum Σess is given by those λ for which the exponent of vA,B(ξ) is purely imaginary,i.e. κ is real. Moreover, the boundary Σess of the essential spectrum is given by those λfor which the solution of Vξ(ξ) = (Aε)±∞(λ)V (ξ) has a purely imaginary exponent ([12]),where (Aε)±∞ = limξ→±∞Aε(ξ, λ). Since limξ→−∞Aε(ξ, λ) = limξ→+∞A(ξ, λ), the operator(Aε)±∞(λ) is written as (Aε)∞(λ). Then

(Aε)∞(λ) =

0 1 0 0ω∞(λ)1−ν2

η(λ)1−ν2

νχ∞(λ)1−ν2

νη(λ)1−ν2

0 0 0 1νω∞(λ)1−ν2

νη(λ)1−ν2

χ∞(λ)1−ν2

η(λ)1−ν2

, (B.7)

where

ω∞(λ) = λ2 +√

1− ε2γ2 + εαλ,

χ∞(λ) = λ2 + 1J

√1− ε2γ2J2 + εαλ.

(B.8)

Under the assumption that vA,B(ξ) = CA,B eiκξ, the system Vξ(ξ) = (Aε)∞(λ)V (ξ) turns into−κ2CA =

ω∞(λ) + iκη(λ)1− ν2

CA − νχ∞(λ) + iκη(λ)

1− ν2CB,

−κ2CB = νω∞(λ) + iκη(λ)

1− ν2CA +

χ∞(λ) + iκη(λ)1− ν2

CB.(B.9)

For the boundary of the essential spectrum this leads to

Σess ={λ(κ) | (κ2 + ω∞(λ) + iκη(λ))(κ2 + χ∞(λ) + iκη(λ))− (κ2ν)2 = 0, κ ∈ R}

,(B.10)


B.1. The Essential Spectrum 75

where CA and CB are related as

CB = −νω∞(λ) + iκη(λ)

κ2(1− ν2) + χ∞(λ) + iκη(λ)CA, (B.11)

with CA still free to choose. Substitution of λ = µ + icκ√1−c2

for simplification converts therequirement for the boundary of the essential spectrum in

Σess ={

λ(κ) ≡ µ(κ) +icκ√1− c2

∣∣∣∣(

µ2 + εαµ +κ2

1− c2+

√1− ε2γ2

)×

×(

µ2 + εαµ +κ2

1− c2+

1J

√1− ε2γ2J2

)−

(εκ2S

1− c2

)2

= 0, κ ∈ R}

.(B.12)

For the case that S = 0 and J = 1 (two uncoupled identical junctions), this results in

µ(κ)1,3 = µ(κ)2,4 = −εα

2±

√ε2α2

4− κ2

1− c2−

√1− ε2γ2

= −εα

2± i

√√1− εγ +

κ2

1− c2− ε2α2

4, for ε ¿ 1,

(B.13)

which corresponds to the boundary of the essential spectrum for a single junction with zerosurface resistance as found in [16, chapter 4]. Back to S 6= 0 and J 6= 1 (although S and Jstill might take these values), solving the equation for µ gives

µ(κ)1,2 = −12εα +

12

ε2α2 ± 2

√(√1− ε2γ2 − 1

J

√1− ε2γ2J2

)2

+ 4(

εκ2S

1− c2

)2

+

− 2(

2κ2

1− c2+

√1− ε2γ2 +

1J

√1− ε2γ2J2

)]12 = −1

2εα +

12

√w±,

µ(κ)3,4 = −12εα− 1

2

ε2α2 ± 2

√(√1− ε2γ2 − 1

J

√1− ε2γ2J2

)2

+ 4(

εκ2S

1− c2

)2

+

− 2(

2κ2

1− c2+

√1− ε2γ2 +

1J

√1− ε2γ2J2

)]12 = −1

2εα− 1

2

√w±.

Since every complex number z can be written as z = |z| ei arg(z) = |z|(cos(arg(z))+i sin(arg(z)))where −π < arg(z) ≤ π, it follows that <(

√z) = <((|z| ei arg(z))1/2) =

√|z| cos(1

2 arg(z)) ≥ 0for z ∈ C. Therefore, <(µ(κ)3,4) ≤ −1

2εα ≤ 0∀ J . Furthermore,√

1− ε2γ2J2 ∈ R for J ≤ 1εγ

and so w± ∈ R with

w± = (2µ(κ)1,2 + εα)2 =

ε2α2 ± 2

√(√1− ε2γ2 − 1

J

√1− ε2γ2J2

)2

+ 4(

εκ2S

1− c2

)2

+

− 2(

2κ2

1− c2+

√1− ε2γ2 +

1J

√1− ε2γ2J2

)

≤ ε2α2 + 2|√

1− ε2γ2 − 1J

√1− ε2γ2J2|+ 4

∣∣∣∣εκ2S

1− c2

∣∣∣∣− 4κ2

1− c2+

− 2√

1− ε2γ2 − 21J

√1− ε2γ2J2

= ε2α2 − 4min(√

1− ε2γ2,1J

√1− ε2γ2J2

)− 4

(κ2

1− c2−

∣∣∣∣εSκ2

1− c2

∣∣∣∣)≤ ε2α2,

(B.14)


June 2005


for J ≤ 1εγ . From this, it follows that <(µ(κ)1,2) ≤ 0 for J ≤ 1

εγ . However, for J > 1εγ , w± have

a non-zero imaginary part. So, <(µ(κ)1,2) > 0 if <(√

w±) > εα. By rewriting the real partof the square root of an imaginary number as <(

√x + iy) = 4

√x2 + y2 cos(1

2 arctan( yx)) =

12

√2√

x +√

x2 + y2, x, y ∈ R, it follows that <(√

w±) > εα if <(w±) + |w±| > 2ε2α2.Because w± are quadratic in κ, they have an extremum at κ = 0, which is a maximum. Forκ = 0, this results in

<(w+) = ε2α2 + 2√

1− ε2γ2 − 2√

1− ε2γ2 = ε2α2,

=(w+) = −21J

√ε2γ2J2 − 1− 2

1J

√ε2γ2J2 − 1 = − 4

J

√ε2γ2J2 − 1.

(B.15)

Now, since =(w+) 6= 0 it holds that <(w+) + |w+| = <(w+) +√<(w+)2 + =(w+)2 >

<(w+) + |<(w+)| = 2ε2α2 and thus <(w+) + |w+| > 2ε2α2. So, if J > 1εγ at least a part of

the µ(κ)1-curve (given by −12εα + 1

2

√w+) lies in the right half-plane. This gives rise to an

instability. However, for J > 1εγ , the fixed points ϕB

2n = arcsin(εγJ) + 2nπ do not exists sothese values for J are actually not relevant.

Consider the region of J ≤ 1εγ , i.e. =(w±) = 0. For this region, (B.14) holds. So,

w± = ε2α2 ± 2

√(√1− ε2γ2 − 1

J

√1− ε2γ2J2

)2

+ 4(

εκ2S

1− c2

)2

+

− 2(

2κ2

1− c2+

√1− ε2γ2 +

1J

√1− ε2γ2J2

)≤ ε2α2, w± ∈ R.

(B.16)

Since µ(κ)1,3 = −12εα± 1

2

√w+ and µ(κ)2,4 = −1

2εα± 12

√w−, this results in

<(µ(κ)1,3) =

−12εα, 0 < J ≤ J∗, ∀κ ∈ R,

−12εα± 1

2

√w+, J∗ < J ≤ 1

εγ, 0 ≤ |κ| < κ∗(J),

−12εα, J∗ < J ≤ 1

εγ, |κ| ≥ κ∗(J),

(B.17)

where J∗ = 4

ε√

16γ2+ε2α4is the maximum value of J for which w+ ≤ 0 for all κ. If J is larger

than J∗, κ∗(J) is the minimum value of |κ| for which w+ ≤ 0. For the imaginary part, itfollows that

=(µ(κ)1,3) =

±12

√−w+, 0 < J ≤ J∗ ∀κ ∈ R,

0 J∗ < J ≤ 1εγ

, 0 ≤ |κ| < κ∗(J)

,±12

√−w+, J∗ < J ≤ 1

εγ, |κ| ≥ κ∗(J).

(B.18)

Furthermore, at κ = 0, it holds that w− = ε2α2− 4max(√

1− ε2γ2, 1J

√1− ε2γ2J2) < 0. So,

the real and imaginary part of µ(κ)2,4 are then given by

<(µ(κ)2,4) = −12εα, 0 < J ≤ 1

εγ, ∀κ ∈ R,

=(µ(κ)2,4) = ±12

√−w−, 0 < J ≤ 1

εγ, ∀κ ∈ R.

(B.19)


B.1. The Essential Spectrum 77

Figure B.1 The real part of µ(κ)1 (peak upwards) and µ(κ)3 (peak downwards) for ε =0.1, γ = 1, α = 1, S = −2 and J ∈ [0, 10].

Making the transformation back to λ (λi = µi + icκ√1−c2

) shows that

<(λ(κ)i) = <(µ(κ)i), i = 1, 2, 3, 4,

=(λ(κ)i) = =(µ(κ)i) +cκ√

1− c2, i = 1, 2, 3, 4. (B.20)

For ε = 0, the boundary of the essential spectrum given by λ(κ)i, i = 1, 2, 3, 4, κ ∈ R iscompletely on the imaginary axis. For ε > 0, the boundary of the essential shifts into the lefthalf-plane and every λ(κ)i is a straight line with <(λ(κ)i) = − εα

2 , i = 1, 2, 3, 4 for J < J∗.


June 2005


(a) λ(κ)1 and λ(κ)3 in the complexplane

(b) λ(κ)2 and λ(κ)4 in the complexplane

Figure B.2 The boundary of the essential spectrum for ε = 0.1, γ = 1, α = 1, J = 2, S = −2and κ ∈ R.

Figure B.3 The imaginary parts of λ(κ)i for ε = 0.1, γ = 1, α = 1, J = 1, S = −2 andκ ∈ R.


B.2. The Point Spectrum 79

B.2 The Point Spectrum

For the point spectrum the values of λ are considered for which there exist a solution tothe eigenvalue problem ∂

∂ξV (ξ) = A(ξ, λ)V (ξ) which also satisfy the boundary conditionslimξ→±∞ V (ξ) = 0. The eigenvalue problem is given by

vA

vAξ

vB

vBξ

ξ

=

0 1 0 0ω(ξ,λ)1−ν2

η(λ)1−ν2

νχ(ξ,λ)1−ν2

νη(λ)1−ν2

0 0 0 1νω(ξ,λ)1−ν2

νη(λ)1−ν2

χ(ξ,λ)1−ν2

η(λ)1−ν2

︸︷︷︸= Aε(ξ, λ)

vA

vAξ

vB

vBξ

, (B.21)

where


χ(ξ, λ) = λ2 + εαλ + cos(ϕBε (ξ))J ,

η(λ) = −c(2λ+εα)√1−c2

,

ν = ε S1−c2

(B.22)

If the system is observed without perturbation, i.e. ε = 0, the eigenvalue problem turns into

Vξ =[

AA0 (ξ, λ) O2

O2 AB0 (ξ, λ)

]V. (B.23)

So, the problem is a combination of the two decoupled systems

(VA)ξ(ξ, λ, 0) = AA0 (ξ, λ)VA(ξ, λ, 0)whereAA

0 (ξ, λ) =

[0 1

λ2 + cos(ϕA0 (ξ)) −2cλ√

1−c2

](B.24)

and

(VB)ξ(ξ, λ, 0) = AB0 (ξ, λ)VB(ξ, λ, 0)whereAB

0 (ξ, λ) =

[0 1

λ2 + cos(ϕB0 (ξ))J

−2cλ√1−c2

].(B.25)

First, the system (B.24) for VA(ξ, λ, 0) is analyzed. The eigenfunction VA(ξ, λ, 0) shouldsatisfy the boundary conditions at ξ → ±∞. Since the operator AA

0 (ξ, λ) is a constantoperator in the limit ξ → ±∞, this eigenfunction VA(ξ, λ, 0) will be an intersection of theunstable manifold and the stable manifold of the asymptotic states. Therefore, the eigenvaluesof limξ→±∞AA

0 (ξ, λ) are considered. From limξ→±∞ cos(ϕA0 (ξ)) = 1, now follows that the

eigenvalues κA1,2 of AA

0 (ξ, λ)|ξ=±∞ are given by

κA1,2 =

−cλ√1− c2

±√

λ2 + 1− c2

√1− c2

= σA1 ± σA

2 . (B.26)

So, by writing λ as λ< + iλ= now for the real part of σA2 holds

<(σA2 ) =

√−λ2

= + 1− c2

√1− c2

, λ< = 0, |λ=| <√

1− c2

0, λ< = 0, |λ=| ≥√

1− c2

(B.27)

and thus <(κA1,2) = ±<(σA

2 ) for λ< = 0. If λ< 6= 0, by using the relation <(√

x + iy) =12

√2√

x +√

x2 + y2, for the real part of σA2 ≡ √

1− c2σA2 =

√λ2< − λ2

= + 1− c2 + 2iλ<λ=,it follows


June 2005


• if |λ=| <√

1− c2 define −λ2= + 1− c2 ≡ δ > 0, then

<(σA2 ) =

12

√2

√λ2< + δ +

√(λ2< + δ)2 + 4λ2

<λ2=

>12

√2

√λ2< + δ +

√(λ2< + δ)2 >

12

√2√

2λ2< = |λ<|.

(B.28)

• if |λ=| ≥√

1− c2 define −λ2= + 1− c2 ≡ −δ ≤ 0, then

<(σA2 ) =

12

√2

√λ2< − δ +

√(λ2< − δ)2 + 4λ2

<(δ + 1− c2)

>12

√2

√λ2< − δ +

√(λ2< − δ)2 + 4λ2

<δ =12

√2

√λ2< − δ +

√(λ2< + δ)2

=12

√2√

λ2< − δ + λ2

< + δ =12

√2√

2λ2< = |λ<|.

(B.29)

Now, <(κA1,2) = −cλ<±<(σA

2 )√1−c2

for λ< 6= 0. So,

<(κA1 ) =

−cλ< + <(σA2 )√

1− c2>−cλ< + |λ<|√

1− c2> 0, λ< 6= 0, ∀λ=,

<(κA2 ) =

−cλ< −<(σA2 )√

1− c2<−cλ< − |λ<|√

1− c2< 0, λ< 6= 0, ∀λ=,

(B.30)

since |c| < 1. This implies that for λ ∈ ΛA0 ≡ C\{ z | <(z) = 0∧|=(z)| ≥ √

1− c2 }, one of theeigenvalues of AA

0 (ξ, λ)|ξ→±∞ has a positive real part and the other eigenvalue has a negativereal part.

Let the eigenfunction V +A (ξ, λ, 0) be the exponential decaying solution (for ξ →∞) of (B.24)

for λ ∈ ΛA0 which corresponds to the eigenvalue κA

2 . In the same way, let V −A (ξ, λ, 0) be the

exponential decaying solution (for ξ → −∞) of (B.24) which corresponds to the eigenvalueκA

1 . Then, there exists a solution VA(ξ, λ, 0) to problem (B.24) if the stable (V +A (ξ, λ, 0))

and the unstable (V −A (ξ, λ, 0)) manifolds intersect for certain values of λ. In ([7],[16]) these

exponential decaying solutions V ±A (ξ, λ, 0) are defined and given by

V +A (ξ, λ, 0) =

e(σA1 −σA

2 )ξ

σA2 + 1

[σA

2 + tanh(ξ)(σA

1 − σA2 )(σA

2 + tanh(ξ)) + sech2(ξ)

], (B.31)

V −A (ξ, λ, 0) =

e(σA1 +σA

2 )ξ

σA2 + 1

[σA

2 − tanh(ξ)(σA

1 + σA2 )(σA

2 − tanh(ξ))− sech2(ξ)

]. (B.32)

Here, V ±A (ξ, λ, 0) are defined in such a way that for λ = 0 they coincide with the eigenfunction

for λ = 0, i.e. the derivative of travelling wave solution (up to a constant). Furthermore,the solutions V ±

A (ξ, λ, 0) are normalized under the conditions limξ→∞

e−(σA1 −σA

2 )ξV +A (ξ, λ, 0) =

[1, σA1 − σA

2 ]T and limξ→−∞

e−(σA1 +σA

2 )ξV −A (ξ, λ, 0) = [1, σA

1 + σA2 ]T, such that they match the

eigenfunctions at ξ = ±∞. Now, the Evans function for ε = 0 is defined as

DA(λ, 0) = e−∫ ξ0 tr(AA

0 (s,λ)) ds(V +A ∧ V −

A )(ξ, λ, 0), (B.33)

where∫ ξ

0tr(AA

0 (s, λ)) ds =−2cλξ√1− c2

= 2σA1 ξ (= (κA

1 + κA2 )ξ ), (B.34)


B.2. The Point Spectrum 81

and

V +A ∧ V −

A = 2σA2

σA2 − 1

σA2 + 1

e2σA1 ξ. (B.35)

Thus, the Evans function is explicitly given by

DA(λ, 0) = e−2σA1 ξ 2σA

2

σA2 − 1

σA2 + 1

e2σA1 ξ = 2σA

2

σA2 − 1

σA2 + 1

=2σA

2 ((σA2 )2 − 1)

(σA2 + 1)2

= 2√

λ2 + 1− c2

√1− c2

λ2

1− c2

1(σA

2 + 1)2.

(B.36)

For values of λ for which the Evans functions equals 0, the stable and unstable manifoldintersect. For that situation, the eigenfunction which solves the eigenvalue problem (B.24)and also satisfies the boundary conditions limξ→±∞ VA(ξ, λ, 0) = 0 is given by the intersectionof V +

A (ξ, λ, 0) and V −A (ξ, λ, 0) for those particularly values of λ. Solving DA(λ, 0) = 0 for λ

results in the double solution λ = 0 and λ = ±i√

1− c2. However, for ±i√

1− c2, the eigen-functions V ±

A (ξ, λ, 0) do not satisfy the boundary conditions since ±i√

1− c2 ∈ (ΛA0 )C . As a

matter of fact, these values lie at (an endpoint of) the boundary of the essential spectrum forε = 0.

Next, the system (B.25) for VB(ξ, λ, 0) is analyzed. At ξ = ±∞, the eigenvalues of AB0 (ξ, λ)

∣∣ξ=∞

are given by κB1,2 = σB

1 ± σB2 , where

σB1 =

−cλ√1− c2

and σB2 =

√λ2 + 1−c2

J√1− c2

. (B.37)

Again, the real parts of the eigenvalues κB1,2 can be determined in a similar way as has been

done for κA1,2. This leads to

<(κB1,2) =

±

√−λ2

= + 1−c2

J√1− c2

, λ< = 0, |λ=| <√

1− c2

J,

0 λ< = 0, |λ=| ≥√

1− c2

J,

(B.38)

and

<(κB1 ) =

−cλ<√1− c2

+ <(σB2 ) >

−cλ< + |λ<|√1− c2

> 0, λ< 6= 0, ∀λ=,

<(κB2 ) =

−cλ<√1− c2

−<(σB2 ) <

−cλ< − |λ<|√1− c2

< 0, λ< 6= 0, ∀λ=,(B.39)

for |c| < 1. Define ΛB0 ≡ C \ { z | <(z) = 0∧ |=(z)| ≥

√1−c2

J }. Then for λ ∈ ΛB0 , the real part

of κB1 is positive and the real part of κB

2 is negative.

Since ϕB(0) = 0 for the [1|0] state, the matrix AB0 (ξ, λ, 0) is a constant matrix. The ex-

ponential decaying solutions V +B (ξ, λ, 0) for ξ → ∞ and V −

B (ξ, λ, 0) for ξ → −∞ are theneasily defined as

V +B (ξ, λ, 0) = e(σB

1 −σB2 )ξ

[1

(σB1 − σB

2 )

], (B.40)

V −B (ξ, λ, 0) = e(σB

1 +σB2 )ξ

[1

(σB1 + σB

2 )

]. (B.41)


June 2005


Now, the Evans function is given by

DB(λ, 0) = e−∫ ξ0 tr(AB

0 (s,λ)) ds(V +B ∧ V −

B )(ξ, λ, 0), (B.42)

where∫ ξ

0tr(AB

0 (s, λ)) ds =−2cλξ√1− c2

= 2σB1 ξ (B.43)

and

V +B ∧ V −

B = 2σB2 e2σB

1 ξ. (B.44)

Substituting these last two relations in the Evans functions leads to

DB(λ, 0) = e−2σB1 ξ 2σB

2 e2σB1 ξ = 2σB

2 =2√

λ2 + 1−c2

J√1− c2

(B.45)

Solving DB(λ, 0) = 0 for λ results in λ = ±i√

1−c2

J . However, for these values of λ, the

eigenfunctions V ±B (ξ, λ, 0) do not satisfy the boundary conditions since ±i

√1−c2

J ∈ (ΛB0 )C ,

i.e. κB1,2 are purely imaginary for these values of λ. Again, these λ’s lie at (an endpoint of)

the boundary of the essential spectrum.

Next, the complete system is considered. The solutions V ±A,B(ξ, λ, 0) of the decoupled systems

are extended as

V ±1 (ξ, λ, 0) =

(V ±A (ξ, λ, 0))1

(V ±A (ξ, λ, 0))2

00

and V ±

2 (ξ, λ, 0) =

00

(V ±B (ξ, λ, 0))1

(V ±B (ξ, λ, 0))2

. (B.46)

In this way, V +1,2(ξ, λ, 0) and V −

1,2(ξ, λ, 0) are eigenfunctions of (B.23). For λ ∈ Λ0 ≡ ΛA0

⋃ΛB

0 =

C\{

z

∣∣∣∣<(z) = 0 ∧ |=(z)| ≥ min(√

1− c2,√

1−c2

J

) }, the eigenfunctions V ±

1,2(ξ, λ, 0) are ex-

ponentially decaying solutions for ξ → ±∞, respectively. The Evans function for the completesystem is now defined as follows

D(λ, 0) = e−∫ ξ0 tr(A0(s,λ)) ds(V +

1 ∧ V +2 ∧ V −

1 ∧ V −2 )(ξ, λ, 0), (B.47)

where∫ ξ

0tr(A0(s, λ)) ds = (2σA

1 + 2σB1 )ξ (B.48)

andV +

1 ∧ V +2 ∧ V −

1 ∧ V −2 = det([V +

1 V +2 V −

1 V −2 ])

= 4σA2 σB

2

σA2 − 1

σA2 + 1

e2(σA1 +σA

2 )ξ . (B.49)

Thus, the Evans function is explicitly given by

D(λ, 0) = 4σA2 σB

2

σA2 − 1

σA2 + 1

=4σA

2 σB2 ((σA

2 )2 − 1)(σA

2 + 1)2

= 4

√λ2 + 1− c2

√λ2 + 1−c2

J λ2

(1− c2)21

(σA2 + 1)2

. (B.50)

Solving D(λ, 0) = 0 for λ results in the double solution λ = 0, λ = ±i√

1− c2 and λ =

±i√

1−c2

J , where the latter four values of λ lie in (Λ0)C , i.e. ±i√

1− c2,±i√

1−c2

J ∈ Σess.


B.3. The Evans function near λ = 0 83

B.3 The Evans function near λ = 0

When ε 6= 0 and λ = 0, the linearized system is given by

V εξ (ξ, 0, ε) = Aε(ξ, 0)V ε(ξ, 0, ε), (B.51)

where

Aε(ξ, 0) =

0 1 0 0ω(ξ,0)1−ν2

η(0)1−ν2

νχ(ξ,0)1−ν2

νη(0)1−ν2

0 0 0 1νω(ξ,0)1−ν2

νη(0)1−ν2

χ(ξ,0)1−ν2

η(0)1−ν2

,

ω(ξ, 0) = cos(ϕAε (ξ)),

χ(ξ, 0) =cos(ϕB

ε (ξ))J

,

η(0) =−εαc√1− c2

,

ν = εS

1− c2.

(B.52)

For V ε(ξ, 0, ε) of the form V ε(ξ, 0, ε) = [vεA, (vε

A)ξ, vεB, (vε

B)ξ]T , the eigenvalue problem (B.51)

is given by{

(vεA)ξξ = cos(ϕA

ε (ξ))vεA − εαc√

1−c2(vε

A)ξ + ε S1−c2

(vεB)ξξ,

(vεB)ξξ = cos(ϕB

ε (ξ))J vε

B − εαc√1−c2

(vεB)ξ + ε S

1−c2(vε

A)ξξ.(B.53)

Since these equations are simply the derivatives (with respect to ξ) of the equations for thetravelling wave solutions (A.7), they are solved by vε

A = (ϕAε )ξ and vε

B = (ϕBε )ξ. Therefore, one

eigenfunction (V ε)+1 (ξ, 0, ε) of (B.51) which satisfies the boundary condition limξ→∞(V ε)+1 (ξ, 0, ε) =0 and one eigenfunction (V ε)−1 (ξ, 0, ε) which satisfies lim

ξ→−∞(V ε)−1 (ξ, 0, ε) = 0 is defined as

(V ε)+1 (ξ, 0, ε) = (V ε)−1 (ξ, 0, ε) = (Cε)±1

(ϕAε )ξ

(ϕAε )ξξ

(ϕBε )ξ

(ϕBε )ξξ

, (B.54)

where (Cε)±1 is a multiplicative constant. By setting (V ε)±1 (ξ, 0, ε)|ε=0 equal to V ±1 (ξ, λ, 0)|λ=0

from the previous section, it follows that (C0)±1 = 14 . Now, since (V ε)+1 (ξ, 0, ε) and (V ε)−1 (ξ, 0, ε)

are not linearly independent it follows that ((V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2 )(ξ, 0, ε) = 0, al-though (V ε)±2 (ξ, 0, ε) are still undefined eigenfunctions satisfying the boundary conditions at±∞. Using this result for the Evans function at λ = 0 and ε 6= 0 leads to

D(0, ε) = e−∫ ξ0 tr(Aε(s,0)) ds((V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2 )(ξ, 0, ε) = 0. (B.55)

Lemma B.1 Let the equation(

∂

∂ξV

)(ξ, λ) = A(ξ, λ)V (ξ, λ) (B.56)

have four solutions V1(ξ, λ), V2(ξ, λ), V3(ξ, λ) and V4(ξ, λ), for which it holds that limξ→∞

G(ξ, λ) =

G+(λ) exists, where

G(ξ, λ) = e−∫ ξ0 tr(A(s,λ)) ds×

×((

∂

∂λV1

)∧ V2 ∧ V3 ∧ V4

)(ξ, λ).

(B.57)


June 2005


Then G(ξ, λ) can be rewritten to

G(ξ, λ) = G+(λ) +∫ ξ

∞e−

∫ τ0 tr(A(s,λ)) ds×

×((

∂

∂λA

)V1 ∧ V2 ∧ V3 ∧ V4

)(τ, λ) dτ.

(B.58)

Proof Differentiating (B.57) with respect to ξ gives

e∫ ξ0 tr(A(s,λ)) ds ∂

∂ξG(ξ, λ) = − tr(A(ξ, λ))

((∂

∂λV1

)∧ V2 ∧ V3 ∧ V4

)(ξ, λ)

+((

∂2

∂ξ∂λV1

)∧ V2 ∧ V3 ∧ V4

)(ξ, λ) +

((∂

∂λV1

)∧

(∂

∂ξV2

)∧ V3 ∧ V4

)(ξ, λ)

+((

∂

∂λV1

)∧ V2 ∧

(∂

∂ξV3

)∧ V4

)(ξ, λ) +

((∂

∂λV1

)∧ V2 ∧ V3 ∧

(∂

∂ξV4

))(ξ, λ).

(B.59)

Since V1(ξ, λ) satisfies differential equation (B.56), it holds that(

∂2

∂ξ∂λV1

)(ξ, λ) =

((∂

∂λA

)V1

)(ξ, λ) +

(A

(∂

∂λV1

))(ξ, λ). (B.60)

Now, by substitution of this relation in the derivative of G(ξ, λ) and the fact that V2(ξ, λ), V3(ξ, λ)and V4(ξ, λ) also satisfy (B.56), it follows that

e∫ ξ0 tr(A(s,λ)) ds ∂

∂ξG(ξ, λ) = − tr(A(ξ, λ))

((∂

∂λV1

)∧ V2 ∧ V3 ∧ V4

)(ξ, λ)

+((

∂

∂λA

)V1 ∧ V2 ∧ V3 ∧ V4

)(ξ, λ) +

(A

(∂

∂λV1

)∧ V2 ∧ V3 ∧ V4

)(ξ, λ)

+((

∂

∂λV1

)∧AV2 ∧ V3 ∧ V4

)(ξ, λ) +

((∂

∂λV1

)∧ V2 ∧AV3 ∧ V4

)(ξ, λ)

+((

∂

∂λV1

)∧ V2 ∧ V3 ∧AV4

)(ξ, λ) =

((∂

∂λA

)V1 ∧ V2 ∧ V3 ∧ V4

)(ξ, λ).

(B.61)

where is made use of the fact that for any matrix M and vectors U1, U2, U3, U4 it holds that(MU1∧U2∧U3∧U4)+(U1∧MU2∧U3∧U4)+(U1∧U2∧MU3∧U4)+(U1∧U2∧U3∧MU4) =( tr(M)U1) ∧ U2 ∧ U3 ∧ U4. So, the derivative of G(ξ, λ) with respect to ξ is now given by

∂

∂ξG(ξ, λ) = e−

∫ ξ0 tr(A(s,λ)) ds

((∂

∂λA

)V1 ∧ V2 ∧ V3 ∧ V4

)(ξ, λ). (B.62)

The expression (B.58) now follows by integrating from∞ to ξ and using limξ→∞

G(ξ, λ) = G+(λ).

¤

RemarkThe lemma can also be used for situations where limξ→−∞G(ξ, λ) exists and isgiven by G−(λ). The proof is equivalent to the proof of the original form of the Lemma andthe rewritten form of G(ξ, λ) is similar to (B.58), but with G+(λ) replaced with G−(λ) andthe integral will be over τ from −∞ to ξ instead of from ∞ to ξ.

Now, with aid of this lemma, the derivative of the Evans function with respect to λ canbe found. The Evans function for λ, ε 6= 0 is defined as

D(λ, ε) = e−∫ ξ0 tr(Aε(s,λ)) ds((V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2 )(ξ, λ, ε). (B.63)


B.3. The Evans function near λ = 0 85

Differentiation of the Evans functions with respect to λ and evaluation at λ = 0 gives

∂

∂λD(λ, ε)

∣∣∣∣λ=0

= e−∫ ξ0 tr(Aε(s,0)) ds×

×

−

∫ ξ

0

(∂

∂λtr(Aε)

)(s, 0) ds

((V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2

)(ξ, 0, ε)︸︷︷︸

=0

+((

∂

∂λ(V ε)+1

)∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2

)(ξ, 0, ε)

+(

(V ε)+1 ∧(

∂

∂λ(V ε)+2

)∧ (V ε)−1 ∧ (V ε)−2

)(ξ, 0, ε)

︸︷︷︸=0

+(

(V ε)+1 ∧ (V ε)+2 ∧(

∂

∂λ(V ε)−1

)∧ (V ε)−2

)(ξ, 0, ε)

+(

(V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧(

∂

∂λ(V ε)−2

))(ξ, 0, ε)

︸︷︷︸=0

,

(B.64)

where the first, third and fifth turn between the brackets ([...]) vanishes since (V ε)+1 (ξ, 0, ε) =(V ε)−1 (ξ, 0, ε). So,

(∂

∂λD

)(0, ε) = e−

∫ ξ0 tr(Aε(s,0)) ds

[((∂

∂λ(V ε)+1

)∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2

)(ξ, 0, ε)

+((

∂

∂λ(V ε)−1

)∧ (V ε)+1 ∧ (V ε)+2 ∧ (V ε)−2

)(ξ, 0, ε)

]= G+(ξ, 0, ε) + G−(ξ, 0, ε).

(B.65)

Since (V ε)±1,2(ξ, 0, ε) are exponentially decaying for ξ → ±∞, respectively, these eigenfunc-tions behave at ξ = +∞ as

(V ε)+1 (ξ, 0, ε) ∼ eκ1(λ,ε)ξ∣∣∣λ=0

, (V ε)+2 (ξ, 0, ε) ∼ eκ2(λ,ε)ξ∣∣∣λ=0

,

(V ε)−2 (ξ, 0, ε) ∼ eκ4(λ,ε)ξ∣∣∣λ=0

and (V ε)−1 (ξ, 0, ε) ∼ (V ε)+1 (ξ, 0, ε),(B.66)

because (V ε)+1 (ξ, λ, ε) = (V ε)−1 (ξ, λ, ε) at λ = 0. Here, κ1,2(0, ε) are the eigenvalues of(Aε)∞(0) with negative real part and κ3,4(0, ε) are the eigenvalues with positive real part. Fur-thermore, by definition (as a result of the normalization), it holds that lim

ξ→∞e−κ1(λ,ε)ξ(V ε)+1 (ξ, λ, ε) =

v+1 (λ, ε), where v+

1 (λ, ε) is the eigenvector corresponding to the eigenvalue κ1(λ, ε). Differen-tiating this relation with respect to λ and subsequently evaluating it at λ = 0 gives

limξ→∞

− e−κ1(0,ε)ξ(V ε)+1 (ξ, 0, ε)︸︷︷︸

=v+1 (0,ε) for ξ→∞

ξ

(∂

∂λκ1

)(0, ε) + e−κ1(0,ε)ξ

(∂

∂λ(V ε)+1

)(ξ, 0, ε)

=(

∂

∂λv+1

)(0, ε).

(B.67)

Hence, for ξ → +∞ (∂∂λ(V ε)+1

)(ξ, 0, ε) ∼ e(κ1(0,ε)+δ)ξ, where 0 < δ = O(1

ξ ) ¿ 1. Therefore,for ξ at +∞ it follows that

G+(ξ, 0, ε) ∼ e−(κ1+κ2+κ3+κ4)(0,ε) ξ e(κ1(0,ε)+δ)ξ eκ2(0,ε)ξ eκ1(0,ε)ξ eκ4(0,ε)

= e(κ1(0,ε)−κ3(0,ε)+δ)ξ → 0,(B.68)


June 2005


for <(κ1(0, ε)) < 0, <(κ3(0, ε)) > 0, 0 < δ ¿ 1 and ξ → +∞. Similarly at ξ = −∞G−(ξ, 0, ε) ∼ e−(κ1+κ2+κ3+κ4)(0,ε) ξ e(κ3(0,ε)−δ)ξ eκ3(0,ε)ξ eκ2(0,ε)ξ eκ4(0,ε)ξ

= e(−κ1(0,ε)+κ3(0,ε)−δ)ξ → 0,(B.69)

for <(κ1(0, ε)) < 0, <(κ3(0, ε)) > 0, 0 < δ ¿ 1 and ξ → −∞. Now, Lemma B.1 can be usedto rewrite the expression for the derivative of the Evans function with respect to λ (B.65),where for G(ξ, λ) of the lemma, it holds limξ→±∞G(ξ, λ) = G±(λ) = 0. This results in

(∂

∂λD

)(0, ε) =

∫ ξ

∞e−


((∂

∂λAε

)(V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2

)(τ, 0, ε) dτ

+∫ ξ

−∞e−


((∂

∂λAε

)(V ε)−1 ∧ (V ε)+1 ∧ (V ε)+2 ∧ (V ε)−2

)(τ, 0, ε) dτ

=∫ −∞

∞e−


((∂

∂λAε

)(V ε)+1 ∧ (V ε)+2 ∧ (V ε)−1 ∧ (V ε)−2

)(τ, 0, ε) dτ,

(B.70)

where the last equality is the result of the fact that (V ε)+1 (ξ, 0, ε) = (V ε)−1 (ξ, 0, ε). To de-termine the explicit expression of ( ∂

∂λD)(0, ε), the eigenfunctions (V ε)±2 (ξ, 0, ε) should bedetermined first. Making an expansion in ε for these eigenfunctions leads to (V ε)±2 (ξ, 0, ε) =(V 0)±2 (ξ) + ε(V 1)±2 (ξ) + O(ε2) (where every (V i)±2 (ξ) is of the form[(vi

A)±2 , ((viA)±2 )ξ(ξ), (vi

B)±2 , ((viB)±2 )ξ(ξ)]T , i = 0, 1). Substituting this expansion in the eigen-

value problem (V ε)ξ = Aε(ξ, 0)V ε gives

• for terms of O(1){

((v0A)±2 )ξξ(ξ) = cos(ϕA

0 (ξ))(v0A)±2 (ξ),

((v0B)±2 )ξξ(ξ) =

1J

(v0B)±2 (ξ).

(B.71)

Then, (v0A)±2 (ξ) = 0 and (v0

B)±2 (ξ) = e∓ξ/√

J , so that (V ε)±2 (ξ, 0, ε) at ε = 0 equalsV ±

2 (ξ, λ, 0) of the previous section at λ = 0.

• for terms of O(ε) (with (v0A)±2 (ξ) = 0 already substituted):

{((v1

A)±2 )ξξ(ξ) = cos(ϕA0 (ξ))(v1

A)±2 (ξ) + S1−c2

((v0B)±2 )ξξ(ξ),

((v1B)±2 )ξξ(ξ) = 1

J (v1B)±2 (ξ)− αc√

1−c2((v0

B)±2 )ξ(ξ).(B.72)

Now, (V ε)±2 (ξ, 0, ε) are defined as

(V ε)+2 = (Cε)+2

(vεA)+2

((vεA)+2 )ξ

(vεB)+2

((vεB)+2 )ξ

and (V ε)−2 = (Cε)−2

(vεA)−2

((vεA)−2 )ξ

(vεB)−2

((vεB)−2 )ξ

, (B.73)

where (Cε)±2 are multiplicative constants. Expanding all terms involved in the derivative ofthe Evans function as follows

ϕAε = ϕA

0 + εϕA1 + O(ε2),

ϕBε = ϕB

0 + εϕB1 + O(ε2) = εϕB

1 + O(ε2),(vε

A)±2 = (v0A)±2 + ε(v1

A)±2 + O(ε2) = ε(v1A)±2 + O(ε2),

(vεB)±2 = (v0

B)±2 + ε(v1B)±2 + O(ε2) = e∓ξ/

√J + ε(v1

B)±2 + O(ε2),(Cε)±1 = (C0)±1 + ε(C1)±1 + O(ε2) = 1

4 + ε(C1)±1 + O(ε2),(Cε)±2 = (C0)±2 + ε(C1)±2 + O(ε2) = 1 + ε(C1)±2 + O(ε2),

(B.74)


B.4. The Evans function near λ = ±i√

1− c2, ±i√

1−c2

J 87

and obtaining the integrand in terms of O(1) and O(ε), the derivative of the Evans functionwith respect to ε becomes

(∂

∂λD

)(0, ε) =

∫ ∞

−∞

18

εc(−(v1

B)+2 +√

J((v1B)+2 )ξ

)eξ/

√J 1

2∂∂ξ ((ϕA

0 )ξ)2√J(1− c2)

− 18

εc((v1

B)−2 +√

J((v1B)−2 )ξ

)e−ξ/

√J 1

2∂∂ξ ((ϕA

0 )ξ)2√J(1− c2)

− 14

c12

∂∂ξ ((ϕA

0 )ξ)2√J(1− c2)

− 14

εc ∂∂ξ ((ϕA

0 )ξ(ϕA1 )ξ)√

J(1− c2)+

18

εα((ϕA0 )ξ)2√J

− 12

εc2αξ 12

∂∂ξ ((ϕA

0 )ξ)2

(1− c2)√

J

−εc(C1)+1

∂∂ξ ((ϕA

0 )ξ)2√J(1− c2)

− 18

εc(C1)+2∂∂ξ ((ϕA

0 )ξ)2√J(1− c2)

− 18

εc(C1)−2∂∂ξ ((ϕA

0 )ξ)2√J(1− c2)

+ O(ε2)

}dξ.

(B.75)

This integral can be further simplified to

(∂

∂λD

)(0, ε) =

∫ ∞

−∞

18

εc(−(v1

B)+2 +√

J((v1B)+2 )ξ

)eξ/

√J 1

2∂∂ξ ((ϕA

0 )ξ)2√J(1− c2)

−εc

((v1

B)−2 +√

J((v1B)−2 )ξ

)e−ξ/

√J 1

2∂∂ξ ((ϕA

0 )ξ)2√J(1− c2)

dξ +

εα√J

+2εc2α

(1− c2)√

J+ O(ε2),

(B.76)

under the assumption that the integral of the terms of O(ε2) converges. Solving (B.72)2 for(v1

B)±2 with (v0B)±2 (ξ) = e∓ξ/

√J under the conditions limξ→∞(v1

B)+2 )(ξ) = 0 andlimξ→−∞(v1

B)−2 (ξ) = 0 results in

(v1B)+2 = (C1− 1

2αcξ√1− c2

) e−ξ/√

J , (v1B)−2 = (C2− 1

2αcξ√1− c2

) eξ/√

J , (B.77)

with C1, C2 constants. Now, integrating the two terms left in the integrand of ∂∂λD leads to

(∂

∂λD

)(0, ε) = − εαc2

(1− c2)√

J+

εα√J

+ 2εαc2

(1− c2)√

J+ O(ε2)

=εα

(1− c2)√

J+ O(ε2) > 0, for ε 6= 0.

(B.78)

B.4 The Evans function near λ = ±i√

1− c2, ±i√

1−c2

J

The zeros of the Evans function at λ = ±i√

1− c2 and λ = ±i√

1−c2

J for ε = 0 lie at theend points of the essential spectrum. The essential spectrum moves into the left half-planeat a speed of O(ε). To determine if the zeros of the Evans function at the end points of theEvans function bifurcate out of the essential spectrum into the right half-plane and cause aninstability, the derivative of the Evans function with respect to ε near the end points of theessential spectrum is analyzed. These end points of the essential spectrum correspond to thevalues of λ for which the operator (Aε)∞(λ) has a double eigenvalue. Recall that the matrixAε(ξ, λ) is given by

Aε(ξ, λ) =

0 1 0 0ω(ξ,λ)1−ν2

η(λ)1−ν2

νχ(ξ,λ)1−ν2

νη(λ)1−ν2

0 0 0 1νω(ξ,λ)1−ν2

νη(λ)1−ν2

χ(ξ,λ)1−ν2

η(λ)1−ν2

, (B.79)


June 2005


where


χ(ξ, λ) = λ2 + εαλ + cos(ϕBε (ξ))J ,

η(λ) = −c(2λ+εα)√1−c2

,

ν = ε S1−c2

.

(B.80)

Hence, the characteristic polynomial of (Aε)∞(λ) is

p(Aε)∞(λ)(κ) =(

κ2 − η(λ)1− ν

κ− 12

ω∞(λ) + χ∞(λ)1− ν

)×

(κ2 − η(λ)

1 + νκ− 1

2ω∞(λ) + χ∞(λ)

1 + ν

)− 1

4(ω∞(λ)− χ∞(λ))2

1− ν2,

(B.81)

where ω∞(λ) = limξ→∞

ω(ξ, λ) and χ∞(λ) = limξ→∞

χ(ξ, λ).

Equal Junctions

For J = 1, the explicit expressions of λ for which (Aε)∞(λ) has a double eigenvalue can bedetermined and are given by

λ = −12εα±(1) i

√1− c2

√1±(2) ν

1±(2) ν(1− c2)

(√1− ε2γ2 − 1

4ε2α2

), (B.82)

where all four combinations of ±(1) and ±(2) for λ give a double eigenvalue. For ε = 0, theλ’s with ±(1) a plus are the end points of the essential spectrum at i

√1− c2 and the λ’s with

±(1) a minus are the end points at −i√

1− c2. The Evans function will be analyzed for theend point of the essential spectrum near i

√1− c2 with ±(2) a minus (the analysis near the

other end points is similar). Define a new parameter ς as follows,

ς = λ +12εα− λ(ε), (B.83)

where

λ(ε) = i√

1− c2

(√1− ν

1− ν(1− c2)

(√1− ε2γ2 − 1

4ε2α2

)− 1

). (B.84)

Now, this new parameter is used to make the transformations

Aε(ξ, ς) = Aε(ξ, λ) and D(ς, ε) = D(λ, ε), so (Aε)∞(ς) = (Aε)∞(λ). (B.85)

By making this transformations, the end point of the essential spectrum corresponds to thatς such that the operator (Aε)∞(ς) has a double eigenvalue. This double eigenvalue occurs atς0 = i

√1− c2 which is a fixed value for ς independent of ε. Furthermore, this eigenvalue of

(Aε)∞(ς0) of algebraic multiplicity 2 is given by

κ(ε) = − ic

1− ν

√1− ν

1− ν(1− c2)

(√1− ε2γ2 − 1

4ε2α2

). (B.86)

The transformed Evans function is given by

D(ς, ε) = e−∫ ξ0 tr(Aε(s,ς)) ds(V +

1 ∧ V +2 ∧ V −

1 ∧ V −2 )(ξ, ς, ε), (B.87)



1− c2, ±i√

1−c2

J 89

where V ±1,2(ξ, ς, ε) are solutions to the equation Vξ(ξ, ς, ε) = Aε(ξ, ς)V (ξ, ς, ε). Since the deriva-

tive of the (transformed) Evans function with respect to ε needs to be analyzed near the endpoint of the essential spectrum, i.e. at ς = ς0 for ε = 0, the functions V ±

1,2(ξ, ς, ε)∣∣∣(ς,ε)=(ς0,0)

are already known. They are given by the eigenfunctions V ±1,2(ξ, λ, 0)

∣∣∣λ=ς0

, which were deter-

mined for the analysis of the point spectrum of the unperturbed system. These eigenfunctionswere given by

V +1 (ξ, λ, 0) =

e(σA1 −σA

2 )ξ

σA2 + 1

σA2 + tanh(ξ)

(σA1 − σA

2 )(σA2 + tanh(ξ)) + sech(ξ)2

00

,

V −1 (ξ, λ, 0) =

e(σA1 +σA

2 )ξ

σA2 + 1

σA2 − tanh(ξ)

(σA1 + σA

2 )(σA2 − tanh(ξ))− sech(ξ)2

00

,

(B.88)

where σA1 = − cλ√

1− c2and σA

2 =√

λ2 + 1− c2

√1− c2

and

V +2 (ξ, λ, 0) = e(σB

1 −σB2 )ξ

001

σB1 − σB

2

, V −

2 (ξ, λ, 0) = e(σB1 +σB

2 )ξ

001

σB1 + σB

2

, (B.89)

where σB1 = − cλ√

1− c2and σB

2 =

√λ2+ 1−c2

J√1−c2

. Hence, for J = 1 and λ = ς0 = i√

1− c2, the

functions V ±1,2(ξ, ς0, 0) are defined as

V +1 (ξ, ς0, 0) = e−icξ

tanh(ξ)−ic tanh(ξ) + sech(ξ)2

00

,

V −1 (ξ, ς0, 0) = e−icξ

− tanh(ξ)ic tanh(ξ)− sech(ξ)2

00

,

V +2 (ξ, ς0, 0) = e−icξ

001−ic

and V −

2 (ξ, ς0, 0) = e−icξ

001−ic

,

(B.90)

i.e. V +1 (ξ, ς, 0) = −V −

1 (ξ, ς, 0) and V +2 (ξ, ς, 0) = V −

2 (ξ, ς, 0).

The derivative of the (transformed) Evans functions with respect to ε at the end point of the


June 2005


essential spectrum is of the form(∂

∂εD

)(ς0, 0) =

∂

∂ε

(e−

∫ ξ0 tr(Aε(s,ς)) ds

(V +

1 ∧ V +2 ∧ V −

1 ∧ V −2

))(ξ, ς, ε)

∣∣∣∣(ξ,ς,ε)=(ξ,ς0,0)

= e−∫ ξ0 tr(A0(s,ς0)) ds

[−

∫ ξ

0

∂

∂εtr(Aε(s, ς0))

∣∣∣∣ε=0

ds(V +

1 ∧ V +2 ∧ V −

1 ∧ V −2

)(ξ, ς0, 0)

+((

∂

∂εV +

1

)∧ V +

2 ∧ V −1 ∧ V−2

)(ξ, ς0, 0)

+(

V +1 ∧

(∂

∂εV +

2

)∧ V −

1 ∧ V −2

)(ξ, ς0, 0)

+(

V +1 ∧ V +

2 ∧(

∂

∂εV −

1

)∧ V −

2

)(ξ, ς0, 0)

+(

V +1 ∧ V +

2 ∧ V −1 ∧

(∂

∂εV −

2

))(ξ, ς0, 0)

],

(B.91)

where all terms in the derivative vanish because of the relations V +1 (ξ, ς0, 0) = −V −

1 (ξ, ς0, 0)and V +

2 (ξ, ς0, 0) = V −2 (ξ, ς0, 0). Hence

(∂

∂εD

)(ς0, 0) = 0, for J = 1. (B.92)

Unequal Junctions

Next, consider J 6= 1. Now, the end points of the essential spectrum lie at

λ = −12εα± i

√1− c2w1(ε, J) and at λ = −1

2εα± i

√1− c2

Jw2(ε, J), (B.93)

for some w1,2(ε, J) which satisfy w1,2(ε, 1) =√

1∓(2)ν

1∓(2)ν(1−c2)(√

1− ε2γ2 − 14ε2α2) (here w1(ε, J)

and w2(ε, J) correspond to the minus and the plus of ∓(2), respectively) and w1,2(0, J) = 1.Again, the Evans function will be analyzed for the end point near λ = i

√1− c2, i.e. λ =

−12εα± i

√1− c2w1(ε, J)

∣∣∣ε=0

. Now the parameter ς, which is used for the transformation ofthe Evans function, is given by

ς = λ +12εα− λ(ε, J), where λ(ε, J) = i

√1− c2 (w1(ε, J)− 1) . (B.94)

So, (Aε)∞(ς) = limξ→∞

Aε(ξ, ς − 12εα+ λ(ε, J)) has a double eigenvalue at (the of ε independent

point) ς = ς0 = i√

1− c2. Although J 6= 1, the functions V ±1 (ξ, ς, ε) still are related as before

at ς = ς0 and ε = 0, i.e. V +1 (ξ, ς0, 0) = −V −

1 (ξ, ς0, 0). However, the relation V +2 (ξ, ς0, 0) =

V −2 (ξ, ς0, 0) is not valid anymore. The functions V ±

1 (ξ, ς0, 0) are the same function as for thesituation of equal junctions, whereas V ±

2 (ξ, ς0, 0) are now given by

V +2 (ξ, ς0, 0) = e(−ic−

√1/J−1)ξ

001

−ic−√

1J − 1

(B.95)

and

V −2 (ξ, ς0, 0) = e(−ic+

√1/J−1)ξ

001

−ic +√

1J − 1

. (B.96)



1− c2, ±i√

1−c2

J 91

Therefore, not all terms vanish in the expression of the derivative of (transformed) Evansfunction with respect to ε at ς = ς0 and ε = 0. The derivative of D is now given by

(∂

∂εD

)(ς0, 0) = e−


{((∂

∂εV +

1

)∧ V +

2 ∧ V −1 ∧ V −

2

)(ξ, ς, ε) +

+((

∂

∂εV −

1

)∧ V +

1 ∧ V +2 ∧ V −

2

)(ξ, ς, ε)

}∣∣∣∣(ξ,ς,ε)=(ξ,ς0,0)

= G+(ξ, ς0, 0) + G−(ξ, ς0, 0).

(B.97)

This expression can be rewritten by using Lemma B.1 to

(∂

∂εD

)(ς0, 0) =

=∫ −∞

∞e−


((∂

∂εAε

)V +

1 ∧ V +2 ∧ V −

1 ∧ V −2

)(τ, ς0, 0) dτ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0).

(B.98)

For ς = ς0 and ε = 0, the eigenfunctions V ±1,2(ξ, ς0, ε) are explicitly known as well as the

expression of Aε(ξ, ς0), i.e.

Aε(ξ, ς0) =

0 1 0 0ω(ξ, ς0)1− ν2

η(ς0)1− ν2

νχ(ξ, ς0)1− ν2

νη(ς0)1− ν2

0 0 0 1νω(ξ, ς0)1− ν2

νη(ς0)1− ν2

χ(ξ, ς0)1− ν2

η(ς0)1− ν2

, (B.99)

where

ω(ξ, ς0) = −14ε2α2 − (1− c2)w1(ε, J)2 + cos(ϕA

ε (ξ)),

χ(ξ, ς0) = −14ε2α2 − (1− c2)w1(ε, J)2 +

1J

cos(ϕBε (ξ)),

η(ς0) = −2icw1(ε, J) and ν = εS

1− c2.

(B.100)

Hence, tr(A0(ξ, ς0)) = −4ic (note that w1(0, J) = 1, according to the definition at thebeginning of this section) and

(∂

∂εAε

)(ξ, ς0)

∣∣∣∣ε=0

= ∂εA0(ξ, ς0) = (B.101)

0 0 0 0−2(1− c2)( ∂

∂εw1)(0, J)− sin(ϕA

0 (ξ))ϕA1 (ξ)

−2ic( ∂∂εw1)(0, J)

S1−c2

{−(1− c2)+ 1

J cos(ϕB0 (ξ))}

−2icS1−c2

0 0 0 0S

1−c2{−(1− c2)+ cos(ϕA

0 (ξ))}−2icS1−c2

−2(1− c2)( ∂∂εw1)(0, J)

− 1J sin(ϕB

0 (ξ))ϕB1 (ξ)

−2ic( ∂∂εw1)(0, J)

.

Using that ϕB0 (ξ) = 0, the derivative of the (transformed) Evans function with respect to ε


June 2005


at the end point of the essential spectrum is given by(

∂

∂εD

)(ς0, 0) =

=∫ −∞

∞det

0 0 − tanh(ξ) 0

(∂εA0(ξ, ς0) eicξV +1 )2 0

−ic tanh(ξ)− sech(ξ)2

0

0 1 0 1

(∂εA0(ξ, ς0) eicξV +1 )4

−√

1J − 1− ic

0

√1J − 1− ic

dξ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0)

=∫ −∞

∞−2 tanh(ξ)

√1J− 1 (∂εA0(ξ, ς0) eicξV +

1 (ξ, ς0, 0))2 dξ + limξ→∞

G+(ξ, ς0, 0)

+ limξ→−∞

G−(ξ, ς0, 0).

Hence, evaluating the expression (∂εA0(ξ, ς0) eicξV +1 (ξ, ς0, 0))2 leads to

(∂

∂εD

)(ς0, 0) =

∫ −∞

∞2

√1J− 1

tanh(ξ)2 sin(ϕA0 (ξ))ϕA

1 (ξ)︸︷︷︸1st term

+

+(

∂

∂εw1

)(0, J)

(2 tanh(ξ)2 + 2ic tanh(ξ) sech(ξ)2

)︸︷︷︸

2nd term

dξ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0).

(B.102)

(Note that for J = 1 the integrand is zero, if limξ→±∞G±(ξ, ς0, 0) is indeed zero, which again

shows that(

∂∂εD

)(ς0, 0) equals 0 for J = 1.) The first term in the integrand is an odd

function in ξ, since tanh(ξ)2 and ϕA1 (ξ) are odd (see Section 2.4.1) whereas sin(ϕA

0 (ξ)) is odd.Hence, this term vanishes by integrating ξ over the whole R. However, the second term is thesum of an even (tanh(ξ)2) and an odd (tanh(ξ) sech(ξ)2) function. Therefore, the resultingexpression of the derivative of the Evans function is

(∂

∂εD

)(ς0, 0) =

∫ −∞

∞4

√1J− 1 tanh(ξ)2

(∂

∂εw1

)(0, J) dξ

+ limξ→∞

G+(ξ, ς0, 0) + limξ→−∞

G−(ξ, ς0, 0).(B.103)


93

Appendix C

AUTO Program

In this appendix, the AUTO files used for obtaining the numerical results by the AUTOsoftware package ([8]) are displayed. The files demonstrated here are especially used forcalculating the solution branch which contains the spiral, as shown in Figure 4.5(a), andsubsequently for obtaining the solution branch containing the spiral inside the closed loop, asshown in Figure 4.5(b).

C.1 The AUTO Equations File (fname.c)

#include "auto_f2c.h"

extern struct {integer itwist, istart, iequib, nfixed, npsi,

nunstab, nstab;}

blhom_1;

These first three lines are required by the AUTO program. Also the following (user-supplied)subroutine func is required by the AUTO program. Inside this subroutine are the differentialequations of the system defined.

int func (integer ndim, const doublereal *u, const integer *icp,

const doublereal *par, integer ijac, doublereal *f, doublereal *dfdu, doublereal *dfdp)

{

doublereal phiA, phiAxi, phiB, phiBxi;

doublereal gammaA, alpha, c, epsilon1, delta1, epsilon2, delta2, epsilon3, delta3, epsilon4, delta4;

doublereal J, S, K;

phiA = u[0];

phiAxi = u[1];

phiB = u[2];

phiBxi = u[3];

gammaA = par[0];

alpha = par[1];

c = par[2];

epsilon1 = par[3];

delta1 = par[4];

epsilon2 = par[5];

delta2 = par[6];

epsilon3 = par[7];

delta3 = par[8];

epsilon4 = par[9];

delta4 = par[11];

J = par[12];

S = par[13];

K = 1./((1.-c*c)*(1.-c*c)-S*S);

f[0] = par[10]*phiAxi;

f[1] = par[10]*K*((1.-c*c)*(sin(phiA) - alpha*c*phiAxi) + S*sin(phiB)/J -

alpha*c*S*phiBxi - (1.-c*c+S)*gammaA);


June 2005

94 C. AUTO Program

f[2] = par[10]*phiBxi;

f[3] = par[10]*K*(S*sin(phiA) - alpha*c*S*phiAxi - (1.-c*c+S)*gammaA + (1.-c*c)*(sin(phiB)/J -

alpha*c*phiBxi));

return 0;

}

The system of differential equations must be defined as the system of first order differentialequations, i.e. the system

{ΦA

xx − ΦAtt = sin(ΦA)− αΦA

t + SΦBxx − γ

ΦBxx − ΦB

tt =1J

sin(ΦB)− αΦBt + SΦA

xx − γ(C.1)

is rewritten to a first order differential equation of ΦA,B(x, t) = ϕA,B(ξ) in ξ, where thetravelling wave coordinate ξ is defined as x − ct, for convenience. Hence, the first ordersystem of differential equations is given by

d

dξϕA(ξ) = (ϕA)ξ(ξ)

d

dξ(ϕA)ξ(ξ) = (1−c2)(sin(ϕA(ξ))−αc(ϕA)ξ(ξ))+S

Jsin(ϕB(ξ))

(1−c2)2−S2

+ −αcS(ϕB)ξ(ξ)−(1−c2+S)γ

(1−c2)2−S2

d

dξϕB(ξ) = (ϕB)ξ(ξ)

d

dξ(ϕB)ξ(ξ) = S sin(ϕA(ξ))−αcS(ϕA)ξ(ξ)−(1−c2+S)γ

(1−c2)2−S2

+ (1−c2)( 1J

sin(ϕB(ξ))−αc(ϕB)ξ(ξ))

(1−c2)2−S2 .

(C.2)

So, the program defined solution vector [u[0],u[1],u[2],u[3]] of the differential equationsis assigned to the variables [phiA,phiAxi,phiB,phiBxi]. Furthermore, the program definedright hand side vector [f[0],f[1],f[2],f[3]], can be defined in terms of phiA,phiAxi,phiBand phiBxi. All the other variables (i.e. γ, α, c, J and S) in the equations are related to entriesin the program defined parameter vector par. Also the parameters epsilon1,delta1,epsilon2,delta2,epsilon3,delta3,epsilon4and delta4 are related to entries in the parameter vector par of the program (the meaningof these parameters will follow later on). At last, the parameter par[10] is used to scale thespatial coordinate, since the program computes the solution on the interval [0, 1].

Next, the initial values of the variables are defined in the (user-supplied) subroutine stpnt.

int stpnt (integer ndim, doublereal t, doublereal *u, doublereal

*par, integer i)

{

doublereal gammaA, alpha, c, J, S, Pi;

doublereal epsilon1, delta1, epsilon2, delta2, epsilon3, delta3, epsilon4, delta4;

par[0] = 0.01; // gammaA

par[1] = 0.18; // alpha

par[2] = -.04359175478; // c

par[3] = 1.0e-03; // epsilon1

par[4] = 1.0e-03; // delta1

par[7] = 0.; // epsilon3

par[8] = 0.; // delta3

par[10] = 1.0e-04; // time-scaling

par[12] = 2.; // J

par[13] = 0.; // S

gammaA = par[0];

alpha = par[1];

c = par[2];


C.1. The AUTO Equations File (fname.c) 95

epsilon1 = par[3];

delta1 = par[4];

epsilon2 = par[5];

delta2 = par[6];

epsilon3 = par[7];

delta3 = par[8];

epsilon4 = par[9];

delta4 = par[11];

J = par[12];

S = par[13];

Pi = 4.*atan(1.);

/* --------------------------------- Initial point ------------------------------------------------ */

u[0] = asin(gammaA) + epsilon1;

u[1] = delta1;

u[2] = asin(gammaA*J) + epsilon3;

u[3] = delta3;

/*--------------------------------------------------------------------------------------------------*/

if (t==1.0)

{

par[5] = 2.*Pi + asin(gammaA) - u[0];

par[6] = u[1];

par[9] = asin(gammaA*J) - u[2];

par[11] = u[3];

}

return 0;

}

In this subroutine stpnt, first the initial values of the variables are assigned to the parametervector par of the program. After that, these parameter vector entries are assigned to thevariables such that they are available for further use inside the function. Also, the numericvalue of π(= 4 arctan(1)) is assigned to the variable Pi. Then, an initial value is assignedto the solution vector [u[0],u[1],u[2],u[3]]. The program uses this assigned initial valuesas the values on time t = 0 (of the interval [0, 1]). Since in the analytical system this in-terval [0, 1] corresponds to [−∞,∞], the values that are assigned to [u[0],u[1],u[2],u[3]]are values that are close to limξ→−∞[ϕA(ξ), (ϕA)ξ(ξ), ϕB(ξ), (ϕB)ξ(ξ)]. To do this, the vari-ables epsilon1,delta1,epsilon3 and delta3 are used which are taken small, i.e. u[0] =limξ→−∞ ϕA(ξ) + epsilon1, u[1] = limξ→−∞(ϕA)ξ(ξ) + delta1, u[2] = limξ→−∞ ϕB(ξ) +epsilon3 and u[3] = limξ→−∞(ϕB)ξ(ξ) + delta3. Furthermore, at t = 1 the program isordered to calculate the parameters epsilon2,delta2,epsilon4 and delta4, which denotethe difference between the solution vector and the asymptotic values of the analytical solu-tion at ξ = ∞, i.e. epsilon2 = par[5] = limξ→∞ ϕA(ξ) - u[0] |t=1, delta2 = par[6] =limξ→∞(ϕA)ξ(ξ) - u[1] |t=1, epsilon4 = par[9] = limξ→∞ ϕB(ξ) - u[2] |t=1 and delta4 =par[11] = limξ→∞(ϕB)ξ(ξ) - u[3] |t=1.

The following subroutine pvls of the program is not necessary in order for the program tocompute solutions and hence it is left empty. After that, in the subroutine bcnd are theboundary conditions defined.

int pvls (integer ndim, const doublereal *u, doublereal *par) {

return 0;

}

int bcnd (integer ndim, const doublereal *par, const integer *icp,

integer nbc, const doublereal *u0, const doublereal *u1, integer ijac, doublereal *fb,

doublereal *dbc)

{

doublereal gammaA, alpha, c, J, S, gammaB;

doublereal epsilon1, delta1, epsilon2, delta2, epsilon3, delta3, epsilon4, delta4;

doublereal Pi;


June 2005

96 C. AUTO Program

gammaA = par[0];

alpha = par[1];

c = par[2];

epsilon1 = par[3];

delta1 = par[4];

epsilon2 = par[5];

delta2 = par[6];

epsilon3 = par[7];

delta3 = par[8];

epsilon4 = par[9];

delta4 = par[11];

J = par[12];

S = par[13];

Pi = 4.*atan(1.);

fb[0] = u0[0] - asin(gammaA) - epsilon1;

fb[1] = u0[1] - delta1;

fb[2] = u0[2] - asin(gammaA*J) - epsilon3;

fb[3] = u0[3] - delta3;

fb[4] = 2.*Pi + asin(gammaA) - u1[0] - epsilon2;

fb[5] = u1[1] - delta2;

fb[6] = asin(gammaA*J) - u1[2] - epsilon4;

fb[7] = u1[3] - delta4;

return 0;

}

In the subroutine bcnd are the boundary conditions of the system defined. This is done byassigning relations to the vector [fb[0],...,fb[7]] which should equal 0, by making use of theprogram’s vectors [u0[0],u0[1],u0[2],u0[3]] (i.e. the values of [u[0],u[1],u[2],u[3]]at t = 0) and [u1[0],u1[1],u1[2],u1[3]] (i.e. the values of [u[0],u[1],u[2],u[3]] att = 1). Hence, the first four relations which are assigned to fb[0],fb[1],fb[2] and fb[3]correspond to the boundary conditions at ξ = −∞, i.e. fb[0] = u0[0] - epsilon1 -limξ→−∞ ϕA(ξ), fb[1] = u0[1] - delta1 - limξ→−∞(ϕA)ξ(ξ), fb[2] = u0[2] - epsilon3- limξ→−∞ ϕB(ξ) and fb[3] = u0[3] - delta3 - limξ→−∞(ϕA)ξ(ξ). Likewise, the last fourrelations which are assigned to fb[4],fb[5],fb[6] and fb[7] correspond to the boundaryconditions at ξ = ∞, i.e. fb[4] = u1[0] - epsilon2 - limξ→∞ ϕA(ξ), fb[5] = u1[1] -delta2 - limξ→∞(ϕA)ξ(ξ), fb[6] = u1[2] - epsilon4 - limξ→∞ ϕB(ξ) and fb[7] = u1[3] -delta4 - limξ→∞(ϕB)ξ(ξ).

The next subroutine icnd defines any integral conditions that are imposed to the systemwhereas the last subroutine fopt is (again) not necessary in order for the program to computesolutions.

int icnd (integer ndim, const doublereal *par, const integer *icp,

integer nint, const doublereal *u, const doublereal *uold,

const doublereal *udot, const doublereal *upold, integer ijac,

doublereal *fi, doublereal *dint)

{

fi[0] = (u[0]-uold[0])*u[1];

return 0;

}

int fopt (integer ndim, const doublereal *u, const integer *icp,

const doublereal *par, integer ijac,

doublereal *fs, doublereal *dfdu, doublereal *dfdp)

{

return 0;

}


C.2. The AUTO Constants File (c.fname) 97

As in the subroutine bcnd the relations that are assigned to entries of fb should equal zero,also in the subroutine icnd, the integral from 0 to 1 over the relations that are assigned toentries of fi should equal zero. Here, the relation assigned to fi[0] is used to avoid thetranslation invariance of the computed solutions.Assume that uk−1(t) is an already computed solution of the system for some combinationof parameter values. Hence, let u(t) be a consecutive solution for the next combination ofparameter values along the solution branch. Then also u(t + σ) is a solution of the system,for any σ. Therefore, for two consecutive solutions it is required that the distance betweenthem is minimized with respect to the translation σ, i.e. the desired solution u(t) is given bythe solution u(t + σ) which minimizes g(σ),

g(σ) =∫ 1

0‖ u(t + σ)− uk−1(t) ‖2

2 dt. (C.3)

Let σ∗ denote the zero of the derivative of g(σ). Then, define the desired solution u(t) asu(t + σ∗) ≡ u(t) and substitute this relation in the derivative of g(σ), i.e.

d

dσg(σ)

∣∣∣∣σ∗

=∫ 1

02(u(t + σ∗)− uk−1(t))u′(t + σ∗) dt =

∫ 1

02(u(t)− uk−1(t))u′(t) dt = 0.(C.4)

Hence, by defining that u(t) is given by the first component of the solution vector[u[0],u[1],u[2],u[3]] which needs to be computed, it follows that u′(t) is given by u[1],whereas uk−1(t) is given by the first component of the last already computed solution vector,i.e. the first component of the program defined vector [uold[0],uold[1],uold[2],uold[3]].In this way, the integral condition which needs to be used, to get rid of the translationinvariance, is given by

∫ 10 (u[0]− uold[0])u[1] dt.

C.2 The AUTO Constants File (c.fname)

4 4 0 0 NDIM,IPS,IRS,ILP

5 10 5 6 9 11 NICP,(ICP(I),I=1,NICP)

35 4 1 0 1 0 8 0 NTST,NCOL,IAD,ISP,ISW,IPLT,NBC,NINT

50 -100.0 100.0 -10.0 10.0 NMX,RL0,RL1,A0,A1

2 0 3 10 7 3 0 NPR,MXBF,IID,ITMX,ITNW,NWTN,JAC

1e-07 1e-07 1e-05 EPSL,EPSU,EPSS

1.0e-02 1.0e-07 1.0 1 DS,DSMIN,DSMAX,IADS

0 NTHL,((I,THL(I)),I=1,NTHL)

0 NTHU,((I,THU(I)),I=1,NTHU)

0 NUZR,((I,PAR(I)),I=1,NUZR)

In this AUTO constants file are the program constants defined, that AUTO requires to beable to run. The most important constants are

• NDIM, the dimension of the differential system as specified in the subroutine func in theAUTO equations file. Here, NDIM is 4.

• IPS, the problem type of the differential system. Here, IPS is 4, which corresponds toa boundary value problem.


June 2005

98 C. AUTO Program

• IRS, the label of the solution (saved in the file s.fname) which is used as initial solutionfor the computation. Here, IRS is 0 for the first run, since there are not yet solutionscomputed and hence the specified initial point in the subroutine stpnt of the AUTOequations file should be used.

• NICP, the number of parameters, in the parameter vector par in the AUTO equationsfile, that AUTO is free to vary in order to satisfy the boundary and integral conditions.Here, NICP is 5 for the first run. More generally, for boundary value problems withintegral constraints, the generic number of free parameters is determined by the relationNBC + NINT - NDIM + 1.

• ICP, the array containing NICP elements that denote which entries in the parametervector par are free to vary. Here, par[10],par[5],par[6],par[9],par[11] are freeto vary in the first run. The first mentioned parameter in the array is the so-calledprincipal continuation parameter.

• NTST, the number of mesh intervals used in the discretization.

• NCOL, the number of Gauss collocation points per mesh interval. Here, NCOL is 4, asrecommended by the program designers.

• NBC, the number of boundary conditions as supplied in the subroutine bcnd in theAUTO equations file that AUTO should use. Here, NBC is 8 since there are eightboundary conditions supplied in bcnd which should be satisfied.

• NINT, the number of integral conditions as supplied in the subroutine icnd in the AUTOequations file that AUTO should use. Here, NINT is 0 for the first run, since in the firstrun the translation invariance is used to obtain a (primary) solution of the desired shape.

• NMX, the maximum number of steps to be taken along the solution branch.

• NPR, every NPR steps the output and plotting data is written to the output file fort.8.

• DS, the pseudo-arclength stepsize for the (first attempted) step along the solutionbranch.

• IADS, every IADS steps the pseudo-arclength stepsize is adapted.

• NUZR, the number of specific parameter values in the vector par, at which labelled outputand plotting data is written to the output file fort.8 or the computation is terminated.

• UZR, the array containing NUZR elements that denote at which specific parameter valuein the vector par labelled output and plotting data is written to the output file fort.8or the computation is terminated.

An explanation of the other (not mentioned) constants of the AUTO constants file can befound in the AUTO manual (available online at http://indy.cs.concordia.ca/auto/).The not mentioned values of the constants in the AUTO constants file are recommendedby the program designers, taking from AUTO constant files which are used for previouscomputations on a two-fold stack of junctions or determined by trial and error to see whichvalue gives the best result.


C.3. The AUTO Runs Executive File (fname) 99

C.3 The AUTO Runs Executive File (fname)

Next, the AUTO file with all the computational runs which AUTO should perform is demon-strated. This file is run by changing the directory to the directory where this file, the AUTOequations file and the AUTO constants file resides in and subsequently giving the commandauto fname in the command line.

In this first run, first the directory is cleared from all old data files that are created byprevious runs of the file, by the commands dl(’fname’) and cl(). This deletion of old datafiles is done to prevent that AUTO continues the labelling of solutions from the last computedsolution in the data files on. Next, the AUTO equations file that should be used, togetherwith the AUTO constants file which contains the initial program constants is loaded by thecommand ld(’fname’).

dl(’fname’)

cl()

ld(’fname’)

print("\n FIRST RUN: increase T from 1.0e-04\n")

r()

sv(’fname’)

print("-------------------------------------------------------")

print(" CONTINUATION WITH SOLUTION NO. 17")

print("-------------------------------------------------------")

plot()

wait()

In the first run, the initial point (defined in the subroutine stpnt of the AUTO equations file)is integrated over time, i.e. the initially small taken scaling parameter par[10] (par[10] isinitially 1.0e-04) is increased, until a solution [u[0],u[1],u[2],u[3]] is computed for whichthe first component has the shape of the desired 2π-kink solution and the second componenthas its maximum (more or less) at t = 1/2 in the computational interval t ∈ [0, 1] that AUTOuses.In this first run, the number of entries in the parameter vector par that AUTO is free to vary(verb”NICP”) is 5, as specified in the AUTO constants file. Which variables that AUTOis free to vary is defined by ICP in the AUTO constants file, i.e. AUTO is free to vary thevalues of par[10],par[5],par[6],par[9],par[11], where par[10] is the scaling parame-ter, par[5] (epsilon2) and par[9] (epsilon4) are the differences between the computedfirst (u[0]) & third component (u[2]) of the solution vector at t = 1 and limξ→∞ ϕA(ξ) &limξ→∞ ϕB(ξ), respectively. Likewise, par[6] (delta2) and par[11] (delta4) are the dif-ferences between u[1] & u[3] and limξ→∞(ϕA)ξ(ξ) & limξ→∞(ϕB)ξ(ξ), respectively. Now,the program is run, with the loaded AUTO constants file and the AUTO equations file, bythe command r(). Subsequently, the output of solution data, bifurcation diagram data anddiagnostic messages are saved to the files s.fname,b.fname and d.fname, respectively, bythe command sv(’fname’). Finally, the solution and bifurcation data is plotted by the com-mand plot() and the program is told to wait by the command wait(). The three lines beforeplot() give an on-screen print of the strings between the (". . ."). After investigation of theplotted solution data, the solution with label 17 is chosen to continue the computation with.

In the second run, the coupling parameter S is decreased from 0 to −0.1456.

print("\n SECOND RUN: decrease S from 0 (T=16.80)\n")

cc("IRS",17)

cc("NINT",1)

cc("NICP",6)

cc("ICP",[13,3,5,7,9,0])

cc("NMX",25)

cc("NPR",10)

cc("DS",-1.0e-04)


June 2005

100 C. AUTO Program

cc("NUZR",1)

cc("UZR",[{’PAR index’:-13,’PAR value’:-0.1456}]) r(s=’fname’)

ap(’fname’)

print("-------------------------------------------------------")

print(" CONTINUATION WITH SOLUTIONS NO. 28")

print("-------------------------------------------------------")

plot()

wait()

First, the initial solution (computed in the first run) to start the computation with is setby the command cc("IRS",17). Furthermore, the position of the solution is fixed by set-ting the number of integral conditions to 1, by the command cc("NINT",1). As a result, alsothe number of free parameters is increased by 1 to 6. The free parameters for this run are set topar[13] (S), par[3],par[5],par[7],par[9] (epsilon1,epsilon2,epsilon3,epsilon4) andpar[0] (γ). As soon as the principal continuation parameter par[13] takes the value−0.1456,the computation is terminated, since this is the desired value for S to continue the computa-tions with. This termination is set by the commands cc("NUZR",1) and

cc("UZR",[{’PAR index’:-13,’PAR value’:-0.1456}]). Then, the AUTO equationsfile is run with the already earlier computed solutions, stored in the file s.fname by the com-mand r(s=’fname’). The new output data is appended to the files s.fname,b.fname andd.fname by the command ap(’fname’). Again, the output data is plotted and the compu-tations will be continued with solution 28, since for this solution the value of S is −0.1456.

In the next run (thirda) the parameter epsilon1 is decreased to (approximately) 0.

print("\n THIRD RUN-A: decrease epsilon1 (T=16.80, S=-0.1456)\n")

cc("IRS",28)

cc("ICP",[10,3,5,7,9,0])

cc("NMX",50)

cc("NPR",10)

cc("DS",1.0e-04)

cc("NUZR",1)

cc("UZR",[{’PAR index’:-3,’PAR value’:0}])

r(s=’fname’)

ap(’fname’)

print("-------------------------------------------------------")


print("-------------------------------------------------------")

plot()

wait()

The initial solution for this run is set to 28, by cc("IRS",28"). Furthermore, the freeparameters for this run are set to par[10],par[3],par[5],par[7],par[9] (the period,epsilon1,epsilon2,epsilon3,epsilon4 and par[0] (γ). The computation is terminatedas soon as par[3] (epsilon1) is (approximately) 0, which occurs for the solution with label 31.

Similar to the previous run, in the thirdb run the parameter epsilon2 is decreased to (ap-proximately) 0.

print("\n THIRD RUN-B: decrease epsilon2 (T=18.21, S=-0.1456)\n")

cc("IRS",31)

cc("ICP",[10,5,7,9,4,0])

cc("NMX",50)

cc("NPR",10)

cc("NUZR",1)

cc("UZR",[{’PAR index’:-5,’PAR value’:0}])

r(s=’fname’)

ap(’fname’)

print("-------------------------------------------------------")


print("-------------------------------------------------------")



plot()

wait()

The initial solution is set to 31. The free parameters for this run are set to par[10],par[5],par[7],par[9],par[4] (the period, epsilon2,epsilon3,epsilon4,delta1) and par[0](γ). The computation is terminated as soon as par[5] (epsilon2) is (approximately) 0,which occurs for the solution with label 34.

Next, in the fourth run the period is increased, such that it becomes large.

print("\n FOURTH RUN: increase T from 21.48 (S=-0.1456)\n")

cc("IRS",34)

cc("ICP",[10,7,9,4,6,0])

cc("NMX",100)

cc("NPR",10)

cc("NUZR",0)

r(s=’fname’)

ap(’fname’)

print("-------------------------------------------------------")


print("-------------------------------------------------------")

plot()

wait()

The initial solution is set to 34. The free parameters are set to par[10],par[7],par[9],par[4],par[6] and par[0] (note that par[3] and par[5] are not used anymore, to keepthem (approximately) 0). The computations are continued with solution 44, for which theperiod is approximately 100. Hence, the [0, 1] AUTO interval corresponds approximately toξ ∈ [−50, 50].

Now, in the fifth run, the (desired) c-γ solution branch is computed, i.e. the IV-characteristicfor S = −0.1456 and J = 2.

print("\n FIFTH RUN: increase gamma from 0.01(T=100,S=-0.1456)\n")

cc("IRS",44)

cc("ICP",[0,2,3,5,7,9])

cc("NMX",500)

cc("NPR",20)

cc("DS",1.0e-02)

cc("IADS",0)

cc("ILP",1)

cc("NUZR",0)

r(s=’fname’)

ap(’fname’)

sv(’fname2’)

print("-------------------------------------------------------")


print("-------------------------------------------------------")

plot()

wait()

The initial solution is set to 44 and the free parameters to par[0] (γ), par[2] (c), par[3],par[5],par[7] and par[9] where, although epsilon1,epsilon2,epsilon3 and epsilon4are free to vary, they should remain approximately 0 in order to obtain an useful approxima-tion of the solution. Furthermore, the constant ILP is set to 1, such that also the solutionsat the turning points in the c− γ curve are labelled and written to the output files. Also theconstant IADS is set to 0, i.e. the pseudo-arclength stepsize along the branch is fixed to thevalue of DS. The obtained c− γ curve is shown in Figure C.1 and the data is (also) saved toa new file fname2.

In the sixth run, the coupling parameter S is decreased from −0.1456 to −0.15, for a solutionnear the centre of the spiral, shown in Figure C.1.


June 2005

102 C. AUTO Program

Figure C.1 The c− γ curve, obtained by the fifth run.

print("\n SIXTH RUN: decrease S from -0.1456 (T=100)\n")

cc("IRS",75)

cc("ICP",[13,3,5,7,9,0])

cc("NMX",100)

cc("NPR",10)

cc("DS",-1.0e-04)

cc("IADS",1)

cc("ILP",0)

cc("NUZR",1)

cc("UZR",[{’PAR index’:-13,’PAR value’:-0.15}])

r(s=’fname’)

ap(’fname’)

print("----------------------------------------------------------------------------------------------------------------")

print(" CONTINUATION WITH SOLUTIONS NO. 80")

print("----------------------------------------------------------------------------------------------------------------")

plot()

wait()

The initial solution is set to 75, which is a solution near the centre of the spiral of FigureC.1. The free parameters are set to par[13],par[3],par[5],par[7],par[9] and par[0],such that par[13] (S) can be decreased to −0.15. As soon as par[13] takes this value, thecomputation is terminated, which occurs for solution 80.

Now, the new c− γ curve (on which solution 80 lies) is obtained for S = −0.15.

print("\n SEVENTH-A RUN: increase gamma (S=-0.15)\n")

cc("IRS",80)

cc("ICP",[0,2,3,5,7,9])

cc("NMX",720)

cc("NPR",100)

cc("DS",1.0e-02)

cc("IADS",0)

cc("ILP",1)

cc("NUZR",0)

r(s=’fname’)

ap(’fname’)

sv(’fname3’)

print("\n SEVENTH-B RUN: decrease gamma (S=-0.15)\n")



cc("IRS",80)

cc("NMX",400)

cc("NPR",100)

cc("DS",-1.0e-02)

r(s=’fname’)

ap(’fname’)

ap(’fname3’)

print("-------------------------------------------------------")

print(" CONTINUATION WITH SOLUTION NO. --")

print("-------------------------------------------------------")

plot(’fname3’)

wait()

In the seventha run, the c − γ curve is computed from the (c, γ) value of solution 80 on.The computed c − γ curve is in the inward direction of the spiral. The computed data isappended to the *.fname files, but also in the new files s.fname3,b.fname3 and ”d.fname3”.Subsequently, the c − γ curve is computed in the seventhb run, from the (c, γ) value ofsolution 80 on. This c − γ curve is computed in the other direction (the outward directionof the spiral). This is done by setting the pseudo-arclength stepsize negative to the value ofDS of the seventha run, i.e. DS is set to −1.0e − 02. The computed data is appended to the*.fname3 files, such that the parts of the c − γ curves are combined. These combined dataare plotted, which results in Figure C.2.

Figure C.2 The c − γ curve, obtained by the combined data of the seventha and seventhb

run.


June 2005

105

List of Symbols

AA(ξ), AB(ξ) Coefficients matrix of the homogeneous equations of O(ε) forthe first junction, coefficients matrix of the homogeneous equa-tions of O(ε) for the second junction.

Aε(ξ, λ),(Aε)∞(λ)

Coefficient matrix of the linearized equations, coefficient matrixof the linearization around the asymptotic states.

α Damping coefficient of the quasi-particle tunnelling current(α > 0).

β Damping coefficient of the quasi-particle current along the su-perconductor’s surface (β > 0).

c Velocity of the travelling wave solutions (|c| < 1).c−, c+ Highest Swihart velocity, lowest Swihart velocity.χ(ξ, λ), χ∞(λ) Entry in Aε(ξ, λ), entry in (Aε)∞(λ).D(λ, 0), D(λ, ε) Evans function for the unperturbed eigenvalue problem, Evans

function for the perturbed eigenvalue problem.ET , EB Potential energy in the top superconductor, potential energy in

the bottom superconductor.η(λ) Entry in Aε(ξ, λ) and (Aε)∞(λ).γ, γA, γB Applied bias current, applied bias current of the first junction,

applied bias current of the second junction (|γ|, |γA|, |γB| ≤ 1).HA(ξ), HB(ξ) Vector containing the inhomogeneous terms ofO(ε) in the equa-

tions for the first junction, vector containing the inhomogeneousterms of O(ε) in the equations for the second junction.

~ Planck’s constant (~ = h2π , where h is the original Planck con-

stant).J Ratio of the critical currents (or critical current densities) of

the first and the second junction (J = jAc

jBc

).jAc , jB

c Critical current density of the first junction, critical currentdensity of the second junction.

κ Wave number of the oscillating perturbations (eigenvalue of(Aε)∞(λ)).

λ Time behaviour determining parameter of the perturbations tothe travelling wave solutions.

µ Transformation parameter (µ = λ + icκ√1−c2

).ν Entry in Aε(ξ, λ) and (Aε)∞(λ).ω(ξ, λ), ω∞(λ) Entry in Aε(ξ, λ), entry in (Aε)∞(λ).P s(ξ), P u(ξ) Projection on the stable subspace for ξ ≥ 0, projection on the

unstable subspace for ξ ≤ 0.P u(ξ), P s(ξ) P u(ξ) ≡ I2 − P s(ξ), P s(ξ) ≡ I2 − P u(ξ).p(Aε)∞(λ)(κ) Characteristic polynomial of (Aε)∞(λ).


June 2005

106 List of Symbols

Φ(x, t), ΦA(x, t),ΦB(x, t)

Phase difference between the top and the bottom superconduc-tor, phase difference over the first junction, phase differenceover the second junction.

ϕA(ξ), ϕB(ξ) Phase difference over the first junction as function of the travel-ling wave coordinate, phase difference over the second junctionas function of the travelling wave coordinate.

ϕA2n, ϕB

2n Stable fixed points of the perturbed system of sine-Gordonequations (ϕA

2n = arcsin(εγ) + 2nπ, ϕB2n = arcsin(εγJ) + 2nπ).

ϕA2n+1, ϕB

2n+1 Unstable fixed points of the perturbed system of sine-Gordonequations (ϕA

2n+1 = π − arcsin(εγ) + 2nπ, ϕB2n+1 = π −

arcsin(εγJ) + 2nπ).ϕA

ε (ξ), ϕBε (ξ),

ϕA,Bi (ξ)

Travelling wave solution of the perturbed system in the firstjunction, travelling wave solution of the perturbed system inthe second junction, term of the travelling wave solution in thefirst/second junction of O(εi).

ϕA1

+(ξ), ϕB1

+(ξ) O(ε)-term of the travelling wave solution in the first junctionbounded on R+, O(ε)-term of the travelling wave solution inthe second junction bounded on R+.

ϕA1−(ξ), ϕB

1−(ξ) O(ε)-term of the travelling wave solution in the first junction

bounded on R−, O(ε)-term of the travelling wave solution inthe second junction bounded on R−.

ΨA1 (ξ), ΨB

1 (ξ) Solution vector for the O(ε)-term of the travelling wave solu-tion in the first junction (ΨA

1 (ξ) = [ϕA1 (ξ), (ϕA

1 )ξ(ξ)]T ), solutionvector for the O(ε)-term of the travelling wave solution in thesecond junction (ΨB

1 (ξ) = [ϕB1 (ξ), (ϕB

1 )ξ(ξ)]T ).ΨT (x, t), ΨB(x, t) Complex valued wave function in the top superconductor, com-

plex valued wave function in the bottom superconductor.ρT (x, t), ρB(x, t) Density of electron pairs in the top superconductor, density of

electron pairs in the bottom superconductor.S Coupling parameter of the junctions (S ∈ [−1, 0]).Σess, Σess Essential spectrum, boundary of the essential spectrum.Σp Point spectrum.ς Transformation parameter (ς = λ+1

2εα−λ(ε), for some functionλ(ε)).

T (λ) Spectral problem operator (T (λ) : U → (d/dξ −Aε(ξ, λ))U).ΘT (x, t), ΘB(x, t) Phase of the wave function in the top superconductor, phase of

the wave function in the bottom superconductor.t Time coordinate.uA(ξ, t), uB(ξ, t) Perturbation to the travelling wave solution in the first junc-

tion, perturbation to the travelling wave solution in the secondjunction.

V (ξ) Solution vector to the eigenvalue problem (V (ξ) =[vA(ξ), vA

ξ (ξ), vB(ξ), vBξ (ξ)]T ).

V +1,2(ξ, λ, 0),

(V ε)+1,2(ξ, λ, ε)Exponential decaying eigenfunctions of the unperturbed eigen-value problem for ξ →∞, exponential decaying eigenfunctionsof the perturbed eigenvalue problem for ξ →∞.

V −1,2(ξ, λ, 0),

(V ε)−1,2(ξ, λ, ε)Exponential decaying eigenfunctions of the unperturbed eigen-value problem for ξ → −∞, exponential decaying eigenfunc-tions of the perturbed eigenvalue problem for ξ → −∞.

vA(ξ), vB(ξ) Solutions to the eigenvalue problem.


107

X(ξ) Fundamental matrix solution of the O(ε)-term of the travellingwave solution in the first junction.

x Spatial coordinate along the junctions.ξ Travelling wave coordinate (ξ = x−ct√

1−c2).

Y (ξ) Fundamental matrix solution of the O(ε)-term of the travellingwave solution in the second junction.


June 2005

109

List of Figures

1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Various Josephson junctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Three stacks of long Josephson junctions. . . . . . . . . . . . . . . . . . . . . 21.4 The Josephson penetration depth λJ of a Josephson junction. . . . . . . . . . 31.5 A fluxon in a two-fold stack of long Josephson junctions ([1|0] state). . . . . . 41.6 A DC SQUID containing two Josephson junctions. . . . . . . . . . . . . . . . 5

2.1 Flowing currents in the Josephson junction, arising from a (local) increase from0 to 2π in the phase difference between the superconductors. . . . . . . . . . 8

2.2 Phaseportrait of the homogeneous sine-Gordon equation in the (ϕ,ϕξ)-plane. 102.3 The soliton solution of the unperturbed sine-Gordon equation, which corre-

sponds to the fluxon in a Josephson junction. . . . . . . . . . . . . . . . . . . 112.4 ϕA

1 (ξ) (solid line) and (ϕA1 )ξ(ξ) (dotted line) for γ = 1. . . . . . . . . . . . . . 17

2.5 ϕB1 (ξ) (solid line) and (ϕB

1 )ξ(ξ) (dotted line) for γ = 1, α = 1, S = −2 andJ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 The boundary of the essential spectrum for ε = 0.1, γ = 1, α = 1, J = 2, S =−2 and κ ∈ R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1 The IV-characteristic for J = 2, S = −0.2 and α = 0.18. . . . . . . . . . . . . 374.2 The derivatives of two solutions on the left part of the branch of Figure 4.1. . 384.3 Two solutions on the right part of the branch (in region B) of Figure 4.1. . . 394.4 The IV-characteristic for J = 2, S = −0.14 and α = 0.18. . . . . . . . . . . . 404.5 IV-characteristics for J = 2 and α = 0.18. . . . . . . . . . . . . . . . . . . . . 414.6 Solutions in the second junction along the spiral of the IV-characteristic for

J = 2, S = −0.14 and α = 0.18. . . . . . . . . . . . . . . . . . . . . . . . . . . 424.7 IV-characteristics for S = −0.1456 and α = 0.18. . . . . . . . . . . . . . . . . 434.8 IV-characteristics for S = −0.1456 and α = 0.18. . . . . . . . . . . . . . . . . 444.9 The IV-characteristic of a stack of identical junctions for S = −0.1456 and

α = 0.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.10 The IV-characteristic for J = 1, S = −0.1456 and α = 0.18. . . . . . . . . . . 464.11 Enlargements of the upper-left part of IV-characteristics for S = −0.1456 and

α = 0.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.12 The IV-characteristic and the corresponding solutions in the second junction

for S = −0.1456 and α = 0.18. . . . . . . . . . . . . . . . . . . . . . . . . . . 484.13 Enlargement of the most left spiral of the IV-characteristics for S = −0.1456

and α = 0.18. Curves from left to right: J = 1, J = 1.00001, J = 1.000015, J =1.00002 & J = 1.00003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.14 The IV-characteristic for J = 0.995, S = −0.1456 and α = 0.18. . . . . . . . . 50

A.1 The travelling wave solution ϕA0 (ξ) of the unperturbed system. . . . . . . . . 56


June 2005

110 List of Figures

A.2 Plot of the elements of the fundamental matrix solution X(ξ). Solid curve:sech(ξ), dotted curve: d

dξ sech(ξ), dashed curve: 12( sech(ξ) + sinh(ξ)), dashed-

dotted curve: ddξ

12(ξ sech(ξ) + sinh(ξ)). . . . . . . . . . . . . . . . . . . . . . . 59

A.3 ϕA1 (ξ) for γ = −2 (solid curve), γ = −1 (dotted curve), γ = 1 (dashed curve)

and γ = 2 (dashed-dotted curve). . . . . . . . . . . . . . . . . . . . . . . . . . 65A.4 (ϕA

1 )ξ(ξ) for γ = −2 (solid curve) and γ = 1 (dotted curve). . . . . . . . . . . 65A.5 ϕB

1 (ξ) for J = 1, solid curve: γ = 1, α = 1, S = −2, dotted curve: γ = 2, α =1, S = −2, dashed curve: γ = 1, α = 2, S = −2 and dashed-dotted curve:γ = 1, α = 1, S = −3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.6 (ϕB1 )ξ(ξ) for J = 1, solid curve: γ = 1, α = 1, S = −2, dotted curve: γ =

2, α = 1, S = −2, dashed curve: γ = 1, α = 2, S = −2 and dashed-dottedcurve: γ = 1, α = 1, S = −3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

B.1 The real part of µ(κ)1 (peak upwards) and µ(κ)3 (peak downwards) for ε =0.1, γ = 1, α = 1, S = −2 and J ∈ [0, 10]. . . . . . . . . . . . . . . . . . . . . 77

B.2 The boundary of the essential spectrum for ε = 0.1, γ = 1, α = 1, J = 2, S =−2 and κ ∈ R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.3 The imaginary parts of λ(κ)i for ε = 0.1, γ = 1, α = 1, J = 1, S = −2 andκ ∈ R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

C.1 The c− γ curve, obtained by the fifth run. . . . . . . . . . . . . . . . . . . . . 102C.2 The c− γ curve, obtained by the combined data of the seventha and seventhb

run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103


111

Bibliography

[1] Arrowsmith, D.K. & Place, C.M., Dynamical Systems : Differential Equations, Maps andChaotic Behaviour. CRC Press, Boca Raton (Florida), 1998. ISBN 0-412-39080-9.

[2] Barone, A. & Paterno, G., Physics and Applications of the Josephson Effect. John Wiley& Sons Inc., New York, 1982. ISBN 0-471-01469-9.

[3] Berg, J.B. van den; Gils, S.A. van & Visser, T.P.P., Parameter Dependence of Homo-clinic Solutions in a Single Long Josephson Junction. Institute of Physics Publishing:Nonlinearity, Vol. 16 (2003), No. 5, Pg. 707-717.

[4] Brown, D.L. e.a., Computation and Stability of Fluxons in a Singularly Perturbed Sine-Gordon Model of the Josephson Junction. SIAM Journal of Applied Mathematics, Vol.59 (1994), No. 4, Pg. 1048-1066.

[5] Coddington, E.A. & Levinson, N., Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955. ISBN 0-07-011542-7.

[6] Coppel, W.A., Dichotomies in Stability Theory. Springer-Verlag, Berlin, 1978. ISBN 3-540-08536-X.

[7] Derks, G.; Doelman, A.; Gils, S.A. van & Visser, T.P.P., Travelling Waves in a SingularlyPerturbed Sine-Gordon Equation. Physica D: Nonlinear Phenomena, Vol. 180 (2003),Issues 1-2, Pg. 40-70.

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[10] Griffiths, D.J., Introduction to Quantum Mechanics. Prentice Hall Inc., Upper SaddleRiver (New Jersey), 1995. ISBN 0-13-124405-1.

[11] Hale, J.K., Ordinary Differential Equations. Wiley-Interscience, New York, 1969. ISBN0-471-34090-1.

[12] Henry, D., Geomatric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin,1981. ISBN 3-540-10557-3.

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112 Bibliography

[14] Pagano, S. & Barone, A., Josephson Junctions. Superconductor Science and Technology,Vol. 10 (1997), No. 12, Pg. 904-908.

[15] Susanto, H.; Visser, T.P.P. & Gils, S.A. van, On Kink-Dynamics of Stacked-JosephsonJunctions, Proceedings Institut Teknologi Bandung, Vol. 34 (2002), No. 2 & 3, Pg. 325-331.ISSN 0125-9350.

[16] Visser, T.P.P., Modelling and Analysis of Long Josephson Junctions. Twente UniversityPress, Enschede, 2002. ISBN 90-365-1759-1.

Internet Resources

[i] WikipediA, http://www.wikipedia.org/

[ii] HyperPhysics, http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html/


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