SOME MATHEMATICAL PROBLEMS IN THE GINZBURG-LANDAU

THEORY OF SUPERCONDUCTIVITY

Stephen J. Gustafson

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

@Copyright by Stephen J. Gustafson, 1999

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SOME bfATHEMATICAL PROBLETVIS IN THE GINZBURG-LANDAU THEORY OF SUPERCONDUCTIVITY Ph-D. Thesis, 1999 Stephen J. Gustafson Graduate Department of Mathematics University of Toronto

Abstract: In agreement with the Landau theory of phase transitions, a su-

perconductor is described macroscopically by the Ginzburg-Landau equations

(1950). These are nonlinear partial differential equations for a cornplex-valued

function. ii, (the order parameter), and a vector-field -4 (the vector potential).

The equations contain a parame ter. A. which determines whet her t hey describe

a superconductor of the first kind ( A < l), or of the second kind ( A > 1). It

was observed by Abrikosov (1957) that a highly-symmetric family of solutions

known as n-vortices plays a central role in the theory. These solutions are

classified by their integer topological degree, n E Z.

The principal goal of this thesis is to establish the stability properties of the

n-vortex. The stability question is studied for three types of evolution equa-

tions: a gradient flow, a nonlinear wave equation, and a nonlinear Schrodinger

equation. Our main result determines the dependence of the stability of n-

vortices on the topological degree, n, and on the parameter, A. Specifically,

we prove that for X < 1, al1 vortices are stable, while for X > 1, n-vortices

are stable if n = rtl and unstable if jnl 2 2. Previous work on vortex sta-

bility (Taubes (1980), Stuart (1994)) has focussed on the special case X = 1,

in which the Ginzbug-Landau equations reduce to the kst-order Bogomolnyi

equations. In particular, our result resolves a long-standing conjecture: first

rigorously formulated by Jaffe and Taubes (1980).

Acknowledgment s

It is a pleasure to thank my supervisor, Prof. LM. Sigal, for his advice and

encouragement, and for al1 that 1 learned from him.

I am very grateful for all the help 1 received from the niembers of the

administrative and library staff of the math department generally, and from

Ida Bulat especially.

1 extend my thanks also to Megan for her support and patience, and to my

colleagues in the department for niaking my time here more enjoyable.

Fiiially, 1 gratefully acknowledge the financial support of NSERC and the

math depart ment.

Contents

1 Introduction I

2 Superconductivity and the Ginzburg-Landau theory 11

2.1 The Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . 11

2.2 Domains and boundary conditions . . . . . . . . . . . . . . . . . 13

2.3 Flux quantization and topology . . . . . . . . . . . . . . . . . . 14

2.4 Lengh scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Yormal m d superconducting solutions . . . . . . . . . . . . . . 17

. . . . . . . . . . . . . . . 2.6 The normal-superconducting interface 19

2.7 -4brikosov vortices . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.8 Vortex stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Linear stability theory 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Properties of the n-vortex . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Vortex solutions . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 Anestimateon thevortexprofîles . . . . . . . . . . . . . 31

3.3 The linearized operator . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Definition of the linearized operator . . . . . . . . . . . . 32

3.3.2 Symmetry zero-modes . . . . . . . . . . . . . . . . . . . 32

3.3.3 Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Block decomposition . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.1 The decomposition of L(") . . . . . . . . . . . . . . . . . 36

3.1.2 Properties of L:' . . . . . . . . . . . . . . . . . . . . . . 38

3.4.3 Tkanslat ional zero-modes . . . . . . . . . . . . . . . . . . 39

3.5 Stability of the fundamental vortices . . . . . . . . . . . . . . . 39

. . . . . . 3.5.1 Non-negativity of LOI and radial minimization 40

3.5.2 A maximum principle argument . . . . . . . . . . . . . . 41

3.5.3 Poçitivity of LOI . . . . . . . . . . . . . . . . . . . . . . 43

3.5.4 Positivity of LI") . . . . . . . . . . . . . . . . . . . . . . 44

. . . . . . . . . 3.5.5 Completion of stability proof for n = Itl 44

3.6 The critical case, X = 1 . . . . . . . . . . . . . . . . . . . . . . . 45

3.6.1 The kst-order equations . . . . . . . . . . . . . . . . . . 45

3.6.2 First-order linearized operator . . . . . . . . . . . . . . . 16

3.6.3 Zero-modes for X = 1 . . . . . . . . . . . . . . . . . . . . 48

3.7 The (in)stability proof for Ini 2 2 . . . . . . . . . . . . . . . . . 49

. . . . . . . . 3.8 Appendk: vortex solutions are radial minimizers 52

4 Nonlinear dynamic stability 53

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Existence and properties of solutions . . . . . . . . . . . . . . . 60

4.3 The stability argument . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Imposition of orthogonality conditions . . . . . . . . . . . . . . 64

4.5 A dynamic instability result . . . . . . . . . . . . . . . . . . . . 68

4.6 Appendix: variational derivative calculation . . . . . . . . . . . 68

Chapter 1

Introduction

This t hesis is concerned principally wit h the nonlinea partial differential equa-

tions known as the Gintburg-Landau (GL) equations? which describe two func-

t ions

The GL equations take the form

The notation needs some explanation. We define Va4 = V - A (the covari-

ant derivative) and AA = Va4 a * Vaa. By mrlA we mean the scalar function

&A2 - &AL For a scalar function, B, we define czlrl B to be the vector field

(-B2B,i?1B)t. We remark that md2.4 = S A - V(V - A). FinaUy, X > O is a

constant parameter.

The GL equations are the Euler-Lagrange equations for critical points of

the functional

which is known as the Ginzburg-Landau energy functional. In fact. it is easily

shown that

and

The functional E(cL., -4) has its origins in the Ginzburg-Landau theory of su-

perconductivity, a variant of the Landau theory of phase transitions. where it

describes the free energy of a superconducting material. In this setting, w is

an "order parameter" whose role it is CO describe locally whether the marerial

is in the superconducting or normal (non-superconduct ing) st ate: where j$l is near zero, the material is in the normal state, and where 1 drl is near one,

the superconducting state prevails. The magnetic field is given by B = curlA.

The quantity I ~ ( $ v - ~ 0) is known as the supercurrent. Equilibrium states are

stationary points of the energy, and hence solve the GL equations. More de-

tails appear in Chapter 2 which contains an overview of the Ginzburg-Landai?

t heory.

The functional E(+, il) also arises in the context of classical gauge-8eld

theory. It describes a Yang-Mills-Higgs theory on R2 with abelian gauge group

U(1). In this setting, A is interpreted as a comection on the principal U(1)-

bundle IR2 x U(l) , and @ is known as a H2ggs field. This theory, and its

non-abelian generalizations, play a fundamental role in quantum fîeld theory

and elementary particle physics.

The GL equations have been well-studied in the critical case X = 1 for which

the equations reduce to the Çst-order Bogomolnyz equations ([Bo], [T 11, [T?] !

[JT], [BGP]). There has also been a lot of recent work on these equations (and

especially their non-magnetic counterpart) in the limit X -t oo ([BBH], [AB],

[SI). The non-magnetic Ginzburg-Landau equation has also been studied for

finite A ([OSl], [BMR], [LL], [Mi]). The work in this thesis is among the first

work on the GL equations for finite, non-critical X.

-4 central feature of the functional A) (and the GL equations) is its

infinite-dimensional s ymmet ry group. S pecifically, E ( d ~ , A) is invariant under

the transformation

for any smooth y : IR2 -+ R. The mapping (1.4) is known as a gauge transfor-

mation. In addit ion, E(+ , -4) is invariant under coordinate translations, and

under the rotation transformation

for any g E SO(2). Al1 of these statements are very easy to check. The

symmetries of E(& -4) play a fundamental role in our analysis.

Finite energy field configurations (y, A) satisfy the boundary conditions

which leads to the definition of the topologzcu2 degree, deg(,@), of such a con-

figuration:

(R sufficiently large). The degree is related to the phenomenon of flux quan-

tization in superconductivity (see Chapter 2 for more details). Indeed, an

application of Stokes' theorem shows that a hite-energy configuration satis-

fies

The simplest non- trivial solutions of the GL equations are the n-vortices

(n E S) which were first studied by Abrikosov ([A]). They are the unique

equivariant solutions, having the "radially-symmetric" form

where (r. 8) are

&(nl (1) = fn ( ~ ) e ' " ' -P)(x) = n- an(r) 2- r

polar coordinates on IR2. i' = ~ ( - X ~ , X , ) ~ , and

f,, a, : [O, oo) -+ P.

It is easily checked that such codigurations (if they satisfy (1.6)) have topo-

logical degree n.

The n-vortices are considered basic building blocks for solutions of the GL

equations. For superconductors of the second kind ( A > l), it is believed

that the n-vort ices, together with periodic vortex lat tices, exhaust the set

of al1 possible solutions. On bounded domains, solutions are expected to be

approximat e nonlinear superpositions of vort ices .

The existence of solutions of the GL equations of the form (1.7) is now

well-known. The h s t existence proof was given by Plohr ([Pl), who used

techniques of the calculus of variations. A more detailed discussion of the

existence theory and properties of n-vortices appears in Chapter 3. For now,

we simply remark that the n-vortex profiles f,(r) and a&) Mnish at r = O

and increase monotonically toward their asppto t ic value of one as r + m.

A partial result on the uniqueness of solutions of the form (1.7) can be found

in [ABGi].

We now address the remaining problem: that of the stability of the n-

vortex. Whereas previous work has focussed on the special case X = 1,

for which the GL equations reduce to the first-order Bogomolnyi equations

([Bo],[JT],[S]), we deal with the rest of the range, X # 1. It was conjectured

in [JT] that the rtl-vortex is stable for al1 values of A, whereas the higher-

degree (ln1 2 2 ) vortices are stable for X < 1 and unstable for X > 1. In this

t hesis, we set tle this conjecture (see Theorem 1).

The n-vortex is a stat ic solution of several natural dynamic ( time-dependent )

versions of the Ginzburg-Landau equations. In order to discuss the notion

of stability, we introduce these equations. First, we need some notation.

We will denote by u a pair of fields, u = ($, A). We will write (abusing

notation slightly) E(u) in place of E ( I , A). The n-vortex will be denoted

,(n) = (iy("), A(")). We now allow u = (W. A) to depend on a time variable, t 7

aswel lasx: il,:@ x ~ , + C a n d . - l : @ x R I W 1 + R 2 .

CVe begin with the simplest dynamic GL equation. the gradient flow (GF)

for the func t ional E ( w , A) :

This equation arises in the study of vortex dynamics in superconductors ([GE]).

It is sometimes written in the following more generd form. Introduce an

addit ional function

and consider the equations

(&A - VV) = V x V x A + I ~ ( . ~ V ~ ~ W ) . (1.10)

It is easily checked that the equations (1.9)-(1 .IO) are invariant (solutions are

mapped to solutions) under time-dependent gauge transformations of the form

for any smooth y : @ x Rt i I. We can recover the simpler form (1.8) of (GF)

fiom (1.9)-(1.10) via a gauge transformation (1.11) with y = - J' V(s )ds . ÇVe

study the Cauchy problem for equation (1.8) with specified initial data

The energy E(u) is non-increasing along a solution of (CF).

In addition to the dissipative gradient-flow equation, there are two natural

conservative tirne-dependent versions of the GL equations. When the GL

equations are studied in the context of gauge field theories, the appropriate

dynamic equation consists of the Maxwell equations for the electric field E =

&A - V and the magnetic field B = V x A coupled to a nonlinear wave

equation for +. The result, known as the Maxwell-Higgs (MH) (or Abelian-

Higgs) equations is

&(A - VV) = cud24 + 1 r n ( $ ~ - ~ i ~ 1 )

V . (&A - VV) = ~m(,$(d< - iV)$) (1.14)

(see, eg, [JT]). (MH) is also studied as a mode1 in superconductivity ([HI).

We supplement equations ( 1.12)-( 1.14) with specified initial data

The blauwell-Higgs equations are also invariant under the gauge transforma-

tion ( 1.11). Forrnally, solut ions of ( 1-13) conserve the energy functional

One can also couple a wave equation for A to a nonlinear Schrodinger

equat ion for iu:

We will refer to equation (1.16)-(1.17) as the nonlinear Schrodinger equation

(NLS). The Cauchy problem assumes Bven initial data

Solutions of (NLS) formally conserve the hinctional

1 lÀ12 + E($, a). 2

A similar equation appeas in ([PT]).

We make the obvious remark that for time-independent fields $J and A

(and with the gauge choice V = O) al1 of these dynamic equations reduce to

the Ginzburg-Landau equations. The existence theory for t hese equations is

discussed in some detail in Chapter 4.

The basic stability question is this: what is the long-time behaviour of a

solution of (GF), (MH), or (NLS) which begins near the n-vortex utn)? The

answer is complicated by the presence of symmetry. Let G,,, denote the group

composed of the gauge-transformations, translations, and rotations described

above. Since the n-vortex dn) is not invariant with respect to the whole group

G,,, (it b'breaks" the symmetry), it gives rise to a wliole manifold of static

solutions, G,,,u(*). al1 of the same energy. It is the stability of this whole

manifold we must consider. Sornewhat informally (for now), we will say that

the n-vortex is orbitally stable if solutions which start close to the manifold

Gyym~(") (we will measure closeness in the H L norm) remain close to G,,dn)

for al1 times.

Our main result conflrms the conjecture of Jaffe and Taubes:

Theorem 1 If either

2. X 2 1 and n = 3 3

then the n-vortex i s orbitally stable a s a solution of (GF), (MH), or (NLS).

For X > 1 and In1 2 2: the n-vortex i s unstable.

This theorem will be more precisely formulated in Chapter 4, where we will

also provide a proof. The instability part of this theorem properly refers to

linear inst ability (see Theorem 2 to follow) . We will, however, present a result

on dynarnic instability for In1 2 2 and X > 1 (Theorem 6 in Chapter 4). This

dynamic instability result relies on work in [G], in which vortex instability is

established in a large X limit (see also [ABG]).

Previous work on vortex dynamics focuses either on the critical case X = 1

([SI, and [AH], [Ml, [Sa] for "soliton" dynamics in gauge theories), or on the

A + m limit ([NI, [El, [CJ], [LX], [G]).

The orbital stability question is closely related to the question of whether

the n-vortex (which is a critical point of the energy E) is a local minimum

or a saddle point: we shall see that only the minimizers are orbitally stable.

This latter problein calls for a study of the second variational derivative of

£ at the n-vortex, which we denote by £"(dn)). This is a symmetric linear

operator on the Hilbert space LZ(R2; @) @ L"R2; R2) (a space of perturbations

of dn) = (d ( " ) , A("))). We will define EU(,u(")) precisely (and exhibit its form)

in Chapter 3. At fkst sight, we e-xpect that if &"(IL(")) is positive definite,

the n-vortex is a local minimizer, and if Ef ' (u( " ) ) has negative spectrum, it is

a saddle point. These considerations are complicated, however, by the pres-

ence of the symmetry group. The following simple principle plays a crucial

role in our analysis: if T generates a one-parameter subgroup of G,, (i.e.

T E Lie(G,,)), then Tu(") E ker(êt '(~dn))). A formal derivation of this fact

goes as follows. We know &E(dn)) = 0, and hence by symmetry invariance

aU~(esTu(")) = O for s E IR. Thus

by the chain rule for variational derivatives. Since the n-vortex is not invariant

under the full symmetry group Gsw, we are faced with a non-trivial kernel for

E1'(u(")). In other words, E"(dn)) is never positive definite. Precise versions

of these considerations appear in Chapter 3.

Let now Ksym denote the subspace of L2 (p ; @) $ L2(R2; IR2) consisting of

elements of Lie(G,,,) applied to dn). Thus hJ, c ker E" (u (~) ) . We Say that

the n-vortex is linearly stable if

and linearly unstable if EI1(u(")) h a negative spectrum. This notion of stability

corresponds to strict local energy minimization modulo symmetry transforma-

tions.

Armed wit h t hese notions, we can st ate the basic time-independent (linear)

stability result :

Theorem 2 1. For al1 X > 0 , the +l -vortex is linearly stable.

2. For Jnl 2 2, the n-vortex is

linearly stable for X < 1

linearly unstable for h > 1.

Theorem 2 underlies the dynamic st ability /inst ability result of Theorem 1, and

will be proved in Chapter 3.

The rest of the thesis is structured as follows. We begin with a brief review

of superconductivity and the Ginzburg-Landau theory in Chapter 2, focusing

on some of the mat hematical problerns which arise t here, and on the connection

with the theorems on vortex stability described above. Chapter 3 is concerned

with linear stability, and contains a proof of Theorem 2. Chapter 1 deals with

the dynamic stability question, and contains the proof of Theorem 1. Each

chapter can basically be read independently of the others.

Chapter 2

~uperconductivity and the

Ginzburg-Landau theory

This chapter contains a brief review of some mathematical aspects of the

Ginzburg-Landau macroscopic theory of superconductivity. Many more de-

tails may be found in the books [TI, [LLi], [Ts], [dG], for example, and the

review articles [CHO], [DGP], and [RI.

2.1 The Ginzburg-Landau theory

Some materials, when cooled below a certain temperature (known as the crit-

ical temperature, Tc) becorne superconductors. That is, a curent can run

through the material with negligible resistance. They also exhibit the Meiss-

ner effect: when a superconductor is placed in an external magnetic field, the

field is expelled from the bulk of the material. The Ginzburg-Landau theory

is a mathematical mode1 which describes these phenomena.

Ginzburg and Landau proposed their phenomenological theory of super-

conductivity in 1950 ([GL]). It is based on the general Landau theory, the

cornerstone of the theory of phase transitions. The Ginzburg-Landau theory

introduces a cornplex-valued order parameter

(here R is the region of space containing the superconductor), whose role i t

is to indicate locally in space whether the rnaterial is in a superconducting or

normal (non-superconducting) state. The magnetic field, B, is e-xpressed, as

usual, in terms of the vector potential

via B = mrl A. The difference in the actual free energy density and the free

energy density of the normal state is then written as an expansion in powers of

t) and its derivat ives. The result (in noruialized units) is an energy functional

of the form

To describe the two phases (the normal phase with !+1 = O and the supercon-

ductingphase with l$l = l), V is taken to be

where X > O is a material parameter (dm is known as the Ginzburg-Landau

parameter). The couphg of 11, to A occurs through the covariant derivative

VA = V - iA, which leads also to the gauge invaxiance of &:

for y : IR3 + R. Physical quantities such as Iqitj and B = mrlA are, of course,

gauge invariant.

The equilibrium state of the superconducting materiai is described by con-

figurations (+, A) which are stationary points of the functional £. Such critical

points satisfy the Ginzburg-Landau (GL) equations

The currently accepted microscopic theory of superconductivity was pro-

posed by Bardeen, Cooper, and Schrieffer in 1957. Two years later. Gor'kov

([Go]) proved that the Ginzburg-Landau theory occurs as a limiting case of the

(generalized) BCS theory, confirrning its fundamental nature. In the BCS the-

ory, electrons form superconducting bound pairs (Cooper pairs). The squared

modulus of the order parameter in the Ginzburg-Landau theory, j @ 1 2 , repre-

sents the local density of Cooper pairs.

2.2 Domains and boundary conditions

A typical physical problem involves a finite superconductor (Cl C R3 bounded)

placed in an external magnetic field H : R -+ IR3. In this case the energy

functional is modified t O become

Often, the external field is a constant, H = hvo, where h 2 O and ,uo is a unit

vector (this will be the case from now on when an external field is present).

In this case, the cross term S m ~ 1 A - H in EH makes no contribution to the

variat ional equations, so the GL equations are unchanged. Typical boundary

condit ions are

where f i denotes the unit normal to 80.

A common situation is when the fields are approximately constant in one

direction (say the 13 direction), and the external field points in that same

direction (uo = (0,O. 1)). In this case, a two dimensional version of the GL

equations is studied. That is. R c IR2? tb : R + C, '4 : R -t EtZ, and mrlA =

al A2 -di A1 is a scalar. The energy functional and the GL equations t hen retain

the same form, with the proviso that for a scalar B, curlB = (-a& & B ) is

a vector. This leads us to the functional (1.3) and the GL equations (1.1-1.2)

as they are written in the introduction. and is the setting for the bulk of the

work in this thesis.

2.3 Flux quant izat ion and topology

One immediate consequence of the Ginzburg-Landau theory is flux quantiza-

tion. Let the total flux of the magnetic field through fl c IR2 be

Using Stokes' theorem, we may rewrite this as a line integral around the bound-

ary :

Now we suppose that the "supercurrent" J = Im(@7.4~~) vanishes on a R

(this will be the case, for example, when we study finite energy solutions on

the plane RZ). We assume also that the materid is superconducting at the

boundary (I$l bounded away from zero on 8R). Writing $.J = l$lei4 with 4

real, we have J = I+I2(Vd - A) = O and hence (V4 - .4)lan = O. By (2.4),

then,

Since q!~ is the phase. and must change by an integral multipie of 27r as dR is

traversed, we have

where n E S is the topological winding number of lil around do . This is flu

quantization. There is a more general notion of "fluxoid" quantization in the

case J does not vanish on the boundary (see, eg, [Tl).

2.4 Length scales

A more subtle feature of the GL theory is the appearance of length scales. We

consider a finite energy solution of (GL) on the whole plane R2. We would like

to determine the rate of decay of the quantities w = 1 - (y1 and B = curlA

as 1x1 + oo (these quantities must decay for the energy to be finite). Writing

S> = I+(ei%ve obtain from (1.1)

Taking the curl of (1.2) we obtain

We study these equations as 1x1 -+ oo? and hence w, B -t O. We remark that

and so Vd - A + O as 1x1 -+ m. Since mrl(Vq5 - -4) = -B we elcpect Vi - A

to decay with the same rate as B. In the leading order as 1x1 -t oo, (2.7)

becomes

We recall that the Greens function for -A + 1 (the integral kernel of the

operator ( -A + l ) - I ) decays like e - ' " - ~ ~ for lx - y1 large. Since the right side

decays at least as quickly as B itself (by the above comments). the solution of

this equation decays Like emi'l. Turning now to (2.6), we obtain

in the leading order as 1x1 + m. The right side decays like e-*IZ', so the

solution of this equation decays as e-mi"(dx2)1x1. To sumrnarize, our forma1

considerations predict decay of the form

as 1x1 -t 00, where r n ~ = 1 and r n ~ = m i n ( 6 . 2 ) . Physicists associate cor-

responding length scales to these rates of e-xponential decay. The penetration

depth, 1-1 = l/mT = 1, is the typicai length scale for variations in the magnetic

field, B. The coherence length is the typical length scale over which the modu-

lus of the order parameter, J$J varies, and is taken in the physics literature to

be ( = 1 / d . The Ginzburg-Landau parameter, K = JX/2' involves the ratio

of the penetration depth to the coherence length: K = l/&<.

Rigorous results are available. In [JT] it is shown that for any finite energy

solution, and any E > O, there is a constant Ad(€) such chat

(so that the decay is at least as fast as predicted abovej. Their proof is based

on the maximum principle. For the specific n-vortex solution described in the

introduction (1.7), Plohr ([P2]) has determined the precise asymptotics of the

fields to be

as r + m. in agreement with the prediction above.

2.5 Normal and superconducting solutions

There are two trivial solutions to the (GL) equations. The firsto un = (,@,, An)

where

corresponds to the normal state (with constant external field H = hqJ. The

second, ,us = (& ,As ) where

represents t be purely superconducting st ate (which exhibits the Meissner effect :

curlA, = O). The respective energies of these solutions are easily computed:

The value h, = 6 1 2 of h for which EH(un) = EH(ur) merits a name: it is

known as the critical magne tic field.

A glance at the functional (2.3) suggests that when the external field h is

sufficiently large, the global energy minimizer of the energy E is the normal

solution, un. When h is sufficiently small, the superconducting solution, u, is

the rninimizer. -1 basic question is how the phase transition from the normal

state to the superconducting state occurs as h is Lowered. To gain some insight

into this question, we linearize (GL) around the normal solution un, and seek

non-trivial solutions. In order to incorporate the external parameter h into

the GL equations, we rescale, setting A = Alh. The GL equations now read

The normal solution .An = An/h satisfies a r l A n = 1. We can choose &(q, 4 =

(O, r l ) , for example. Writing I,O = and A = .in + Ca, and retaining only

the leading order terms in (2.8) as E i O, we obt ain

Equation (2.10) determines cr once ( is known. Equation (2.9) has a very

simple form: it is a Schrodinger eigenvalue equation for a charged particle in

a constant magnetic field. The solutions are well-known. The ground state

energy (bottom of the spectrum) of -AhAn is h. Thus the highest value of h

for which non-trivial solutions exist is h,, = X/2 (h,, is known as the upper

critical field). The corresponding family of generalized eigenfunctions is

This computation suggests that a superconducting solution can bifurcate

fiom the normal solution at h = h,. We remark that for X < 1, h,, = X/2 <

h, = 6 1 2 . -4s the field is lowered, then, by the time a bifurcation from the

normal solution is possible, the purely superconducting solution is energetically

favourable. In this case. one e-xpects an abrupt phase transition (a phase

transition of the first kind) to the purely superconducting state. Conversely,

if X > 1. a gradua1 transition for h, < h < h,, through a ''rn~xed state" is

possible. We therefore expect superconductors with X < 1 (known as type 1

superconductors), to exhibit qualitatively different properties t han t hose wit h

X > i (type II superconductors).

For a mathematical analysis of some critical magnetic fields? see [BH] and

references t herein.

2.6 The normal-superconducting interface

In [GL], Ginzburg and Landau made a calculation of the "surface energy" of

an interface separating a normal korn a superconducting region. The sign of

this quantity was found (numerically) to be the same as the sign of 1 - A. We

now give a brief description of this calculation.

We consider a one-dimensional solution of the GL equations describing a

transition between the normal and superconducting states. The fields A (real-

valued now) and 1,6 depend on a single variable, x. By a gauge transformation,

we can assume z,b is real-valued. To the far left, we impose the condition that

the material is in the normal state with an external field strength of h, = d / 2 :

as x i - x . To the far right, we impose the normal state:

as x + m. Let ut,,, = (.yt,.,., .l,,.,,) denote a solution of the GL equations.

which now take the form

satisfying the above boundary conditions (such a solution is sometirnes called

a domain wall). Since h = h,, the energy densities of the normal and purely

superconducting states are the same (equal to the constant X/8). If e(utranS)

denotes the free energy density of the solution ut,.,,, then e(ut,,,,) - X/8

vanishes as x -+ km. Set b = j"_",{e(ut,,,) - X/8). Thus

We are interested in the sign of the quantity b as an indicator of the energy

cost (or profit) associated with the formation of a normal-superconducting

interface. To simpli& the expression for b we use the virial relation

obtained by multiplying the Ç s t GL equation (2.12) by and integrating by

parts. This leads easily to \ POO A s = l-_ {(l - ~ ' / h . ) ~ - i4).

The author is unaware of a rigorous analytic evaluation of d. In the physics

literature (eg [TI), heurist ic considerations are used to determine the approx-

imate value of b in the limiting cases X << 1 and X >> 1. In the former

case, 6 > 0, and in the latter, 6 < O. Numerical computations suggest that 6

changes sign exactly at X = 1.

2.7 Abrikosov vortices

The nature of the mived state in a superconductor of the second kind ( A >

1) was studied by Abrikosov in 1957 ([A]). Beginning with the idea that a

normal-superconducting interface carries a positive surface energy for a Type

II superconductor, he predicted the appearance of a periodic array of "flux

tubes,!' for h below h,.

In studying the nature of nucleation of the normal state from the purely

superconducting one as a small external field is increased, Abrikosov predicted

an initially sparse flux lattice? and was led to the problem of an isolated flu~

ce11 in an infinite domain. This was the motivation behind his introduction of

the family of radially-spmetric n-vortex solutions of the form

which were described in the introduction.

For h just below, h,,, the Abrikosov flux lattice is described by a superpo-

sition of the solut ions (see equation 2.11) of the linearized GL equations:

which is periodic in x2 (with period 2ajk). If Cn+N = C,, for sorne integer

N 2 1, then (2.14) is also periodic in X I (with period k N / h ) . It turns out

that N is the winding number of @ around the fundamental cell. The choice

C, = C for al1 n is known (for k = 1/2*h) as .4brikosov's square lattice. Each

ce11 contains a single vortex of degree one. The choice C2, = C, =

is known (for k = &=) as the triangular lattice. Here N = 2, and each

ce11 contains two vortices in such a way that the vortices form a triangular

array. The lowest energy lattice was determined numerically in [KRA] to be

the triangular one.

A few rigorous mathematical results concerning these Iattices are available.

Odeh ([O]) proved the existence of periodic solutions to the full GL equations.

Odeh ([O]) and Milman and Keller ([MK]) studied the bifurcation from the

normal state at h = h,. Almog ([Al]) and Chapman ([Cl) establish the ex-

istence of many periodic solut ions of the linearized equations, and invest igate

the stability (with respect to periodic perturbations) of some of them.

2.8 Vortex stability

Since the n-vortex (2.13) plays a basic role in the Ginzburg-Landau theory,

the question of its stability is a fundamental one. The stability question was

h s t addressed by Bogomolnyi ([Bo]) who provided a non-rigorous argument

suggesting that the n-vortex with In1 2 2 is unstable (not a minimizer) when

h > 1 (type II superconductor). Our proof of the instability part of Theorem 2

(see the introduction) is partially based on Bogomolnyi's fundamental idea.

Bogomolnyi's work also ernphasized the special role played by the transitional

value A = 1.

'a numerical error led Abrikosov to conchde that the square lattice had the minimal

energy.

The solutions of the GL equations are well-understood in the case of critical

couplzng, X = 1. In this case? the Bogomolnyi rnethod ([Bo]) gives a pair of

&st-order equations whose solutions are global minirnizers of & ( a @ , A) among

fields of Lxed degree, and hence solve the GL equations. Taubes ([Tl, T21)

h a . shown that al1 solutions of GL with X = 1 are solutions of these fust-order

equations, and that for a given degree n: the gauge-inequivalent solutions form

a 2 (n l-parameter family. The 2 (n ( parameters describe the locations of the zeros

of the order parameter (il. This is discussed in more detail in [JT] (see also

[BGP]) and Chapter 3. We remark that for X = 1. the n-vortex solution (2.13)

corresponds to the case when ail In1 zeros af w lie at the origin.

There is a somewhat vague general picture concerning inter-vortex inter-

actions. Consider a finite energy configuration u = (,+, A) for which i has

isolated zeros with well-defined local winding numbers. We think of these ze-

ros as localized vortices. The general picture suggests that any pair of vortices

experiences an inter-vortex force whose sign is determined as follows:

1. if the winding numbers have different signs, the force is attractive

2. if the winding numbers have the same sign, the force is

(a) attractive if X < 1

(b) zero if X = 1

(c) repulsive if X > 1

An attractive (resp. repulsive) force means the energy is lowered by decreas-

h g (resp. increasing) the vortex separation. There is some minimal evidence

supporting t his picture. Jacobs and Rebbi ( [JR] ) have performed a numer-

ical calculation in which the energy is first rninimized among configurations

containing two vortices, each of degree +1, a k e d distance R apart. As R

is increased. the energy of the minimizer is found to increase when X < 1,

remain unchanged for X = 1, and decrease for X > 1, as is predicted by the

%de" described above. The Taubes solution for X = 1 is also consistent with

this picture, as it allows arbitrary placement of vortex centres with no energy

change. Indeed, t his general picture under lies the following old conjecture

which is clearly formulated in the 1980 book of Jaffe and Taubes ([JT]): for

X < 1, the n-vortex is stable for all n, whereas for X > 1, only the Il-vortices

are stable, while higher-degree (In1 2 2) vortices are unstable. Our main re-

sult (see Theorem 1 in the introduction) settles this conjecture. One thinks

of the n-vortex (In1 2 2) as a pile of In1 single vortices at the origin. It is

then energetically favourable for these vortices to split apart (instability) only

if X > 1.

Chapter 3

Linear stability theory

3.1 Introduction

In this chapter. we study the linearized stability of Ginzburg-Landau n-vortices.

ive begin by recalling some basic definitions.

The n-vortices are critical points of the energy functional

for the fields

Here VA4 = V - iA is the covariant gradient, and X > O is a coupling constant .

For a vector, A, V x A is the scalar &Az - &Al1 and for a scalar {, V x ( is

the vector ( -&c, ai(). Criticd points of E(,@, A) satis& the Gznzburg-Landau

(GL) equations

where iîa4 = V-4 -V.+

The functional E($, A) arises in the Ginzburg-Landau theory of supercon-

ductivity (see Chapter 2) , as well as in Yang-Mills-Higgs classical gauge theory.

The functional E ( @ , A) (and the GL equations) exhibit an infinite-dimensional

symmetry group. Specifically, E(.iu, A ) is invariant under U ( 1) gauge transfor-

mations,

for any smooth y : lR2 i W. In addition, &($? A) is invxiant under coordinate

translations, and under an additional coordinate rotation transformation which

plays no role here.

Finite energy field configurations satisS,

which leads to the definition of the topological degree, de&@), of such a con-

figuration:

(R sufficiently large). The degree is related to the phenornenon of flux quan-

tization (see Chapter 2) .

We study, in particular, "radially-symmetric" or "equivariant" fields of the

form

where (r, 8) are polar coordinates on IR2, I L = f (-x2, xl)', TZ is an integer, and

Such configurations have topological degree n. The existence of critical points

(solutions of (3.2-3.3)) of this form is well-known (see section 3.7.1). Thep are

called n -vortices.

Our main results concern the stability of these n-vortex solutions. Let

be the second variation of the Eunctional E around the n-vortex. acting on the

space

The symmetry group of &(y, A) gives rise to an infinite-dimensional subspace

of ker(L(")) c r(' (see section 3.3.2), which we denote here by h;,,. We say

the n-vortex is linearly stable if for some c > 0.

and linearly unstable if L(*) has a negative eigenvalue. The basic result of

this paper is the following linearized stability statement (which appears as

Theorem 2 in the introduction):

Theorem 3 1. (Stabilzty of f indamental vortices)

For al1 X > 0, the *l-vortex zs linearly stable.

2. (S ta bQt y/ànstability of higher-degree vortices)

For In1 2 2, the n-vortex is

linearly stable for A < 1

lznearly unstable for A > 1.

Theorem 3 is the basic ingredient in a proof of the nonlinear dynamical

stability/instability of the n-vortex for certain dynamical versions of the GL

equations (see Chapter 4).

The statement of theorem 3 was conjectured in [JT] on the bais of numeri-

cal observations (see [JR]). Bogomolnyi ([Bo]) gave an argument for instability

of vortices for X > 1, In1 2 2. Our result rigorously establishes this property.

The solutions of (3.2-3.3) are well-understood in the case of critical cou-

plzng, ,\ = 1. In this case, the Bogomolnyz method ([Bo]) gives a pair of

first-order equat ions whose solutions are global minimizers of &(&, A) among

fields of fked degree (and hence solutions of the the GL equations). Taubes

([Tl. 721) has shown that dl solutions of GL with X = 1 are solutions of these

h s t -order equat ions, and t hat for a given degree n. the gaugeinequivalent

solutions form a 2 ln 1-parameter family. The 2171 1 parameters describe the lo-

cations of the zeros of the scalar field. This is discussed in more detail in

section 3.6. We remark that for X = 1, an n-vortex solution (3.6) corresponds

to the case when al1 In1 zeros of the scalar field lie at the origin.

We rernark that for this work, the essential features of the nonlinear po-

tential

1 V(I4) = pl2 - oz

appearing in the energy (3.1) are that it exhibits a maximum at = O and

decreases to a non-degenerate minimum on the circle l'@ 1 = 1.

The chapter is organized as follows. In section 3.2 we describe in detail

various properties of the n-vortex. In particdax, we establish an important es-

tirnate on the n-vortex profiles which differentiates between the cases X < 1 and

X > 1. In section 3.3, we introduce the linearized operator, fix the gauge on the

space of perturbations, and identify the zero-modes due to symmetry-breakuig.

Sections 3.4 through 3.7 comprise a proof of theorem 3. A block-decomposition

for the linearized operator is described in section 3.4. This approach is sirnilar

to that used to study the stability of non-magnetic vortices in [OS11 and [Gu].

In section 3.5, we establish the positivity of certain blocks (those correspond-

ing to the radially-symmetric variational problem, and t hose cont aining the

translational zero-modes) for al1 A, which completes the stability proof for

the Il-vortices. The basic techniques are the characterization of symmetry-

breaking in terms of zero-modes of the Hessian (or linearized operator), and a

Perron-Fkobenius type argument, based on a version of the maximum princi-

ple for systems (proposition 6), which shows t hat the translational zero-modes

correspond to the bottom of the spectrum of the linearized operator. A more

careful analysis is needed for In1 2 2. This requires us to review some aspects

of the critical case ( A = 1) in section 3.6. The stability/instability prool for

in1 2 2 is completed in section 3.7. We use an extension of Bogornolnyi's

inst ability argument, and anot her application of the Perron-Fkobenius theory.

3.2 Properties of the n-vortex

In this section we discuss the existence, and properties, of n-vortex solutions.

3.2.1 Vortex solutions

The existence of solutions of the GL equation of the form (1.7) is well-known:

Theorem 4 (Vortex Existence; [P, BC]) For every integer n, t h e ~ e is a

solution

of the GL equations (1.1)-(1.2) of the f o n (1.7). In particular, the radial

finctions (f,, 4) minimize the radial energy functional

do (1 - a)' f 2 ., (a')2 X ( f a) = + no i-n--p- + $ f 2 - 1)'

2 0 ,r2

(which is the full energy finctional (1.3) restricted to fields of the form (1.7))

in the class

The functions f,, a, are smooth, and have the following properties (for n # O ) :

3. f, -. cr", a, -- dr2. as r + O (c > O and d > O are constants)

4. 1 - fn, 1 - a, -t O as r -t oc, wzth an exponential rate of decay.

We cal1 (@("), -4'")) an n-oortes (centred at the origin).

It follows immediately that the functions f , and a, satisfy the ODES

n2(1 - a,) 2 A -&fn + r* fn + ?(f; - 1 ) f n = 0

and

In [ABGiI it is proved that the n-vortex is the unique solution of the form (1.7)

for X 2 2n2. This result is based on the observation that for X in this range,

any such solution minimizes E!"). A proof of this latter fact is given in an

appendiv (section 3.8).

Remark 1 The finctions f, and a, also depend on A, but w e suppress this

dependence for ease of notation. When it wdl cause no confusion, we ,wdl also

drop the subscript n.

3.2.2 An estimate on the vortex profiles

The following inequality, relating the exponentially decaying quantities f' and

1 - a! plays a crucial role in the stability/instability proof.

Proposition 1 We have

Proof: Define e(r ) f ' ( r ) - n(l-ra(r)) f ( T ) . The properties listed in theorem 4

imply that e ( r ) -t O as r i O and as r i oc. Using the ODES ((3.8)-(3.9))

we can derive the equation

where

and the result follows from the maximum principle. O

3.3 The linearized operator

In this section, we introduce the linearized operator (or Hessian of E) around

the n-vortex, and identifjr its symmetry zero-modes.

3.3.1 Definition of the linearized operator

FVe work on the real Hilbert space

x = L ~ ( I R ~ ; cc) ) L*(R*~ R*)

with inner-product

We define the linearized operator, LG,.4 (= the Hessian of E ( Q , A ) ) at a solution

($! A) of (1.1)-(1.2) through the quadratic form

for al1 ( E l B ) , ( q , C), E X. The result is

3.3.2 Symmetry zero-modes

We identib the part of the kernel of the operator

which is due to the symmetry group.

Proposition 2 We have

for any y : IR2 + W

Proof: We use the basic result that the generator of a one-parameter group of

symmetries of E(+ , A). applied to the 12-vortex. lies in the kernel of L("). The

vector in (3.1 1) is easily seen to be the generator of a one-parameter family

of gauge transformations ( 1.4) applied to the n-vortex. Similarly, the vector

in (3.12) is the generator of coordinate translations applied to the n-vortex.

Of course, both (3.11) and (3.12) are easily checked directly. O

Rernark 2 Applying the generator of the coordinate rotational symmetry (1.5)

to the n-uortex giues us nothing new, it is contained in the gauge-syrnmetry

case.

We define Ks,, to be the subspace of X spanned by the L2 zero-modes

described in proposition 2. FVe recall that the n-vortex is called stable if there

is a constant c > O such that

and unstable if L(*) has a negative eigenvalue.

3.3.3 Gauge fixing

In order to remove the infinite dimensional kernel of L(") arising from gauge

symmetry, we restrict the class of perturbations. Specifically, we restrict L*(")

to the space of those perturbations ((: B) E X which are orthogonal to the LZ

gauge zero-modes (3.11). That is,

for al1 y. Integration by parts in this equation gives the gauge condition

As is done in [SI, we consider a modified quadratic form L(") , defined by

for a = (<, B) E X. Clearly, L(") agrees with L(") on the subspace of X speci-

fied by the gauge condition (3.14). This modification has the important effect

of shifting the essential spectrum away from zero (see (3.33)). A straightfor-

ward computation gives the following expression for i("):

To establish theorem 3. it suffices to prove that i(") 2 c > O on the subspace

of X orthogonal to the t ranslat ional zero-modes (3.12).

L(") is a real-linear operator on X. It is convenient to identify L2(R2; P) wit h L2 (R2 ; @) t hrough the correspondence

and then to complexify the space X tt 2 = [L*(R~; @ ) I d via

As a result? E(") is replaced by the cornplex-linear operator

= (n) L = diag {-AA, -=, -A, -A) + vcn)

$(21$4* - 1 ) + f ld~l' - ( i(.';w)

i ( a , @ ) lzo12 -i(d+\@) O

Here we have used the notation

where 8: = al -id2 (and the superscript c has been dropped from the complex

hinction A obtained from the vector-field A via ( 3 . 1 5 ) ) .

The components of v(") are bounded, and it follows from standard results = (n)

([RSII]) that L is a self-adjoint operator on .I, with domain

3.4 Block decomposition

We mite functions on W2 in polar coordinates. Pre

where Lr,, i L2 (Et+, rdr) .

Let p, : U(1) + Aut([LZ(SL; @ ) I d ) be the representation of the complex

unit circle, U ( l ) , whose action is given by

where R, is a counter-clockwise rotation in through the angle a. It is easily = (n)

checked that the linearized operator L commutes with p,(g) for any g E = (n)

U (1). It follows that L leaves invariant the eigenspaces of dp,(s) for any s E = (n)

iP = Lie(U(1)) . The resulting block decomposition of L , which is described

in this section. is essential to our analysis. In particular, the translational

zero-modes each lie within a single subspace of this decomposition.

3.4.1 The decomposition of L ( ~ )

In what follows, we define, for convenience, b ( r ) = 41-47)) r

Proposition 3 The~e is an orthogonal decomposition

=(II) under ~which the linearized operator around the uortez, L . decomposes as

where

with

I - diag {[m + n(1 - a)I2, [m - n(1 - a)12, [m - il2, [m + il2) + v' r *

and

Proof: The decomposition (3.18) of x follows from the usual Fourier decompe

sition of L2(S' ; C), and the relation (3.17). An easy computation shows that =in) L preserves the space of vectors of the form

and that it acts on such vectors via (3.19). O

It follows that lm) is self-adjoint on [L?,].'.

It will also be convenient to work with a rotated version of the operator

where

By a simple computation, we have

where

3.4.2 Properties of L:)

Proposition 4 We hadue the Jollowing:

1.

oess ( L:') = [min( i? A), m)

3. For In[ = 1 and m > 2.

iuith no zero-eigenvalue.

Proof: The first statement is obvious. The second statement follows in a

standard way from the fact that

To prove the third statement, we compute

r n - 1 L:) - Lp) = diag {m + 2741 - a), m - 2 4 1 - a ) , m - 1, rn + 3) r2

which is non-negative, with no zero-eigenvalue for m 2 2, n = 1. D

Remark 3 In light of (3.22). we can assume from now on that rn 1 O . This

double degeneracy 2s a resvlt of the complexifzcation (3.16) of the space of

perturbations.

3.4.3 Translational zero-modes

The gauge fxing (section 3.3.3) has eliminat ed the zero-modes arising honi

gauge symmetry, but the translational zero-modes remain.

As written in (3.12), the translational zero-modes fail to sat isb the gauge

condition (3.14). Further, they do not lie in L2. A straightforward computa-

tion shows that if we adjust the vectors in (3.12) by gauge zero-modes given

by (3.11) with y = -.$, j = 1,%: we obtain

where el = (1, O)' and e2 = (O. 1)'. Tl and T2 satisfy the gauge condition (3.14),

and are zero-modes of the linearized operator. Note also that TkI decay e.xpo-

nentially as lx 1 + oo, and hence lie in L'.

It is easily checked that Tl k iT2 lie in the rn = *1 blocks for L$). After

rotation by R, we have

w her e

3.5 St ability of the fundamental vortices

In this section we prove the tirst part of theorem 3. Specifically, we show

that for some c > O, LE') 2 c for m # 1, and L:*') ITL 2 C. In Light of the

discussions in sections 3.3.3, 3.4.1, and 3.4.3, this will establish the stability of

the & 1-vortices.

3.5.1 Non-negat ivity of and radial rninirnizat ion

Proposition 5 2 O for al1 ,\.

Proof: From the expression (3.21) we see that ~ r ) decomposes:

L F ) = No @ Mo

(abusing notation slightly) where

with

and

An easy computation shows that MO is precisely the Hessian (second variation)

of the radial energy, ~ e s s ~ ? ) (see (3.7)). Since the n-vortex minimizes E!"),

we have ildo 2 O. It remains to show iVo 2 O. We establish the stronger result,

iVo > O. We begin with the observation that

where

In fact, Ga has no zero-eigenvalue. To see this, we first remark that Go is

a relatively compact perturbation of GoI,\=l: due to the exponential decay

of the field components. It follows fiom an index-theoretic calcdation done

in [W], [SI, that Golx=i is Fredholm. with index 0. W e conclude that the same

is true of Go (for any A) . Finally, it is a simple matter to check that Gi has

trivial kernel. if

it follows that

and hence that 3 = 0, and so E = O. The relation Xo > O follows from this,

and the the fact t hat ue,,(No) = [l, m). O

3.5.2 A maximum principle argument

Removing the equality in proposition 5 requires more work. First. we establish

an extension of the maximum principle to systems (see, eg, [LM, Pa] for related

results) . We will use this also in the proof that the the translational zero-mode

is the ground state of ~( i ) (section 3.5.4).

Proposition 6 Let L be a self-adjoint operator o n L'(Rn; Rd) of the f o n

where V i s a d x d mat%-multiplication operator with smooth entries. Suppose

that L 2 O and that for i # j , Kj(x) 5 O for al1 x. Further, suppose V is

irreducible in the sense that for any splitting of the set (1 , . . . , d ) in to disjoint

sets Si and S2, there is an i E SI and a j E S2 with Vj(x) < O for al1 x.

Finally, suppose that L{ = q E L2 ~ i t h q > O component -he , and E f O. Then either

2. q zz O and < < 0.

Proof: We mite [ = - (- with Fi, [- 2 O component-wise, and compute

Since tJ- and (; have disjoint support, we have

Thus we have

2. O = < CJ-, V& > for al1 j f k

Since L 2 0, the first of these relations implies L(- = O and hence LE+ = q.

So if 7 $ 0, then 5' $ O. If s O and Cr t O, replace < with -5 in what

follows. An application of the strong maximum principle (eg. [GT] Thm.

8.19) to each component of the equation

now allows us to conclude that for each k, either {l > O or C = O. We know

that for some k , C > O. Looking back at the second listed equation above,

and using the irreducibility of V, we then see that t3T ,- O for all j. Finally,

we can easily rde out the possibility ck = O for some k , by looking back at the

equation satisfied by &. Thus we have < > O. 0

3.5.3 Positivity of

Now we apply proposition 6 to show Mo > O. The trick here is to find a

function 5 which satisfies ilfoc 2 0. This allows us to rule out the existence of

a zero-eigenvector! which woiild be positive by proposition 6. To obtain such

a (, we differentiate the vortex with respect to the parameter A. Specifically,

differentiation of the Ginzburg-Landau equations with respect to X results in

where

We can now establish

Proposition 7 For al1 A , ~ r ) 2 c > 0.

Proof: We have already shown in the proof of proposition 5, that No > O and

iVIo 2 O. Hence, due to (3.25) and (3 .23) , it suffices to show that Wz~l l ( iV1~) =

(O). Suppose Mo j = O, j f O. Proposition 6 then implies C > O (or else take

-C). Now

gives a contradiction. O

Remark 4 Proposition 6 applied to equation (3.26) also gives E > O . That

is, the ,vortex profiles increase monotonicaZZy .with A. This c m be used to show

that the rescaled .vortex (f, ( r / A), a&/ A)) converges as X -t m to ( f *, O ) ,

where f* is the (profile of) the n-uortex solution of the ordinary GL equation:

-A, f' + n'f * / r 2 + (fa' - 1) f * = O . This result was proved by different rneans

in [.4BGj, and used to establish the instability of higher-degree vortices for large

A.

3.5.4 Positivity of L\")

Proposition 8 L:'~' 2 O with non-degenerate zero-eigenvalue given b.y T .

Proof: Let p = in f specf,lr '' 5 O, which is an eigenvaiue by (3.23). Sup-

pose L(,")s = pS. Applying proposition 6 to LI (=') - p (note that V/ sat-

isfies the irreducibility requirement) gives S > O (or S < O). Further, p is

non-degenerate. as if p were degenerate. we would have two strictly positive

eigenfunctions which are orthogonal, an impossibility. Now if p < O, we have

< S, T >= O, which is also impossible. Thus S is a multiple of T. and p = 0.

O

3.5.5 Completion of stability proof for n = f 1

We are now in a position to complete the proof of the k s t statement of the-

orem 3. By proposition 7, @') 2 c > O. By proposition 8 and (3.23),

L ( ; E ' ) I ~ L 2 E > O. Finally, by (3.24), L:" 2 c' > O for Iml 2 2. It fol-

lows h m proposition 3 that z(") 2 c > O on the subspace of X orthogonal

to the translational zero-modes. By the discussion of section 3.3.3, this gives

theorem 3 for n = kl, [3

3.6 The critical case, X = 1

In order to prove the remainder of theorem 3, we exploit some results from the

X = 1 case.

3.6.1 The first-order equations

Following [Bo], we use an integration by parts to rewrite the energy (1 -3) as

- 1)l2 (3.27)

(recall, since we work in dimension two: V .< A is a scalar) vhere deg(v) is

the topological degree of w , defined in the introduction. We assume, without

loss of generality. that deg(@) 2 0. Clearly, when X = 1. a solution of the

first-order equations

minimizes the energy within a fked topological sector, de&@) = n, and hence

solves the GL equations. Note that we have identified the vector-field A with

a complex field as in (3.15).

The n-vortices (1.7) are solutions of these equations (when X = 1). Specif-

icall y,

and

In fact, it is shown in [T2] that for X = 1, any solution of the variational

equations solves the first- order equations (3.28-3.29).

Beginning from e.upression (3.27) for the energy, the variat ional equations

(previously mitten as (3.2-3.3)) can be written as

(here 81 = -& + iA is the adjoint of &).

3.6.2 First-order linearized operator

We show that the linearized operator at X = 1 is the square of the linearized

operat or for the first-order equations.

Linearizing the first-order equat ions (3.28-3.29) about a solution, (u, .A) (of

the fmt-order equations) results in the following equations for the perturba-

tion, a = (c , B) :

Now using -i& B = V x B - i(V . B ) , and adding in the gauge condition (3.14) ,

we can rewrite this as

where

If we linearize the full (second order) variational equations (in the form (3.32-

3.33)) around (@. .A), we obtain

and

i.6[&< - iB@] + i<[8.4w] - i&[V x B + ~ e ( $ < ) ] = 0.

Proposition 9 When h = 1. these linean'zed equations can also be written

L;L@ = O

Proof: This is a simple computation using the fact that the first-order equa-

tions (3.28-3.29) hold. O

This relation holds also on the level of the blocks. A straightforward corn-

putat ion gives

where

Fm =

3.6.3 Zero-modes for A = 1

It was predicted in [W] (and proved rigorously in [SI) that for X = 1, the

linearized operator around any degree-n solution of the first-order equations

has a 2 lnl-dimensional kernel (moduio gauge transformations). This kernel

arises because the Taubes solutions form a 2lnl-parameter family, and al1 have

the same energy. The zero-eigenvalues are identified in [Bo], and we describe

them here. Let 1, be the unique solution of

m2 (-AT + - + f 2 ) x m = O

r

on (O, oc) with

and

for m = 1 , 2 ,

where

Yrn + O as r + o o

n. Then it is easy to check that

F*,CV,,, = O

We remark that

and it is easily verified that for X = 1, WI1 = T are the translational zero-

modes.

3.7 The (in)stability proof for In1 2 2

Here we complete the proof of theorem 3.

The idea is to decompose L:) into a s u m of two terms, each of which has

the same (translational) zero-mode (for rn = 1) as L:). One term is manifestly

positive, and the other satisfies restrictions of Perron-Frobenius theory.

We begin by modifjring Fm, and defining, for any A,

where we have defined

and a, - q denotes an operator composition. By (3.31): we have q 1 for X = 1.

We also set, for m = 1,. . . : n,

Now I.Y, has the following properties:

1. KV,, is the translational zero-mode T for al1 X

- 2. when X = 1, tVm = W,, rn = f 1,. . . , kn, give the 271 zero-modes (3.35)

of the ünearized operator.

These were chosen in [Bo] as candidates for directions of energy decrease

(for Iml 1 2) when X > 1. Intuitively, we think of Wm as a perturbation that

tends to break the n-vortex into separate vortices of lower degree.

Now, Fm was designed to have the following properties:

- 1. Fm = Fm when X = 1 (this is clear)

- - 2. F,bK = O for al1 rn and X ( t his is easily checked).

A straightforward computation then gives

LK' = F;F, + Ji&,

where J = diag(1, 0,0,0) and

wit h

By construction. when m = 1, the second term in the decomposition (3.37)

must have a zero-mode corresponding to the original translational zero-mode.

In fact, one can easily check that Ml f' = 0.

Proposition 10 For ln( 2 2, Mt has a non-degenerate zero-ezgenvalue corre-

spondzng to f ', and

on Lf,,.

Proof: We recall inequality (3.10), which implies that for X < 1, q < 1, and

for X > 1, q > 1. The operator Mt is of the form

Ml = (1 - q 2 ) (-A,) + first order + multiplication. (3.38)

One can show that MI is bounded from below (resp. above) for X < 1 (resp.

X > 1). We stick with X < 1 for simplicity. The h > 1 case is similar.

Suppose iCIlq = pq with p = in f spec:Lli 5 0, and q f O. Applying the

maximum principle (eg proposition 6 for d = 1) to Mt - pl we conclude that

q > O. If p < O, we have < rl, f ' >= O, a contradiction. Thus p = 0: and is

non-degenerate by a similar argument. C1

We also have

Lemma 1 For rn > 2. .Ilm - Ml is non-negative for X < 1. non-positiue for

X > 1. and has no zero-eigen~value.

Proof: This follows easily h m the equation

Completion of the proof of theorem 3: Suppose now A < 1. Since F,'$,, is manifestly non-negative, and l'LIm > Mi for rn 2 2, we have L$) 2 0 for

m 2 1 (with only the translational O-eigenvalue). Combined with (3.23) and

propositions 7 and 3, this establishes the stability of the n-vortex for X < 1.

Wow suppose X > 1. By (3.37), proposition 10 and lernma 1, we have for

m = k2, ... &n,

We rernark that fi; does, in fact, correspond to an element of the original (un-

complexified) space X, and so L(*) has negative eigenvalues. This establishes

the instability of the n-vortex for In1 2 2, X > 1, and completes the proof of

theorem 3. Cl

3.8 Appendix: vortex solutions are radial min-

imizers

Proposition 11 For X > 2n2, a solution of the equations (3.8-3.9) minzmizes

Proof: It suffices then to show Mo = ~ e s s ~ ! " ) > O (see section 3.5.1). We

mi t e Mo = LLo + Zo where

with 1 = -A, + b2 + :( f 2 - 1) and

We note that 1 f = O (one of the GL equations). It follows from the fact that

f > O and a Perron-Frobenius type argument (see [OS 11) that 1 2 O with no

zero-eigenvalue. It suffices to show Zo 2 O. Clearly tr(Zo) > 0, and

is strictly positive for X 2 2n2. n

Chapter 4

Nonlinear dynamic stability

4.1 Introduction

This chapter is devoted to the study of the time-dependent stability and in-

stability of the Ginzburg-Landau n-vortex. We begin by recalling some basic

defini t ions.

The ( time-independent ) Ginz burg-Landau (GL) equations

for the functions

il : R' -t 3 and $J : IR2 + C (4.3)

are the Euler-Lagrange equations satisfied by critical points of the functional

known as the Ginzburg-Landau energy. Here V-4 = V - iil is the covariant

derivative, AA = Vm4 . Va4, and X > O is a constant. We will often simplify

notation by writing u = ( W . -4) for a pair of functions as in (4.3).

The functional &( I,O, -4) arises in the Ginzburg-Landau t heory of supercon-

ductivity (see Chapter 2), and as an example of a Yang-Mills-Higgs classicd

gauge t heory.

A central feature of the functional E ( $ . -4) (and consequently of the GL

equations) is its infinite-dimensional symmet ry group. Specifically, E(I,!J A) is

invariant under U ( 1 ) gauge transfonnations.

for any smoot h y : IR2 i R. In addition. E ( @ , A) is invariant under coordinate

translations (x it x - b). We denote by G,, the group composed of these

transformations. and for g E G,,, we mi te gu for the result of transforming

the fields u by g.

Finite energy field configurations satisfy

which leads to the definition of the topological degree, deg(,t,b), of such a con-

figuration:

deg(@) = deg - (IL!

(R sufficiently large).

We are concerned with a well-known family of solutions of the GL equations

cailed n-uortices, n E Z, which have the form u(") = (+("), A(")) with

where ( T , 19) are polar coordinates on R2. The radial profiles

vanish at the origin, and tend (exponentially quickly) to 1 as r + m. The

existence of such solutions for any n, A, is established in [PI, [BC]. Further,

there is known to be a unique solution of this form for X 2 4n2 ([ABGi]).

The GL equations (4.1-1.2) have several natural time-dependent general-

izations. The simplest of these is the gradient flow (GF) for the functional

which mises in the theory of superconductivity ([GE]). LVe study the Cauchy

problem for (4.7) wit h specified initial condition

Of course, (GF) is invariant under (time-independent ) gauge transformations (4.5).

In the context of gauge field theory, the appropriate dynamic equations are

the Maxwell-Higgs (MH) (or Abelian-Higgs) equations,

& ( A - VV) + cur12.4 + Irn(i>vA$) = O

V - ( A - V V ) = ~ m ( $ ( & - iV)@).

Here V : @ x Rt + R is the scalar potential, and the electric field is E = -4 -

VV. Equations (4.9)-(4.10) are the Maxwell equations with charge 1rn(,6(& -

il;).+ and current I ~ ~ ( $ V . ~ . O ) . The Cauchy problem for (MH) prescribes initial

data

çat isfying

v (Po - VV0) = I,m(y,,7ro).

Equat ions (1.8)-(4.10) are invariant (solutions map to solut ions) with respect

to time-dependent gauge transformations of the form

for smooth y : IR: x IRt -+ R. We will work fkom now in the temporal gauge,

V O. which can be achieved through a gauge transformation with =

We consider also a nonlinear Schrodinger (NLS) equation for ~, coupled to

a wave equation for -4:

Similar equations appear in [PT] as a mode1 in superconductivity. For (NLS)

we speci& initial data

Equations (4.11)-(4.12) are invariant with respect to t ime-independent gauge

transformations (4.5).

An existence theory has been developed for (4.7) in [DS] and for (4.8)-(4.10)

in [BM]. We will give more details in section 4.2. The author is unaware of

any existence results for (4.11)-(4.12). The n-vortex u(") is a static (time-

independent) solution of al1 of these equations (with V O for (MH)).

The purpose of the work in this chapter is to determine the stability of

the n-vortex as a solution of these evolution equations. In light of the large

symmetry groups of these equations, the n-vortex gives rise to a whole man-

ifold of solutions, and it is the stability of this manifold we must address.

Therefore, the appropriate notion of stability here is t hat of orbital stabilzty.

More precisely, we realize the symmetry group G,, (gauge transformations

and coordinate translations) as H2 x IR2, where (y , b) E H' x IR2 corresponds

to a gauge transformation by y (see (4.5)) and a coordinate translation by b.

Let

be the symmetry orbit of the n-vortex dn). Define the distance of a pair of

fields u = (,@, A) to GSy,,dn) by

dzst (u, G , ~ , U ( ~ ) ) E inf 11 u - gu(nl 11 Lvt. 9 E G , ,

We will Say that the n-vortex is orbitally stable as a solution of one of the

above dynamical equations if given any c > O there is a B > O such that for

any solution u ( t ) with initial data satis&ing

for all t 2 0.

We can now state the main result of the chapter (which appears as Theo-

rem 1 in the introduction):

Theorem 5 If either

then the n-vortex is orbitally stable as a soliltion of (GF) or ( A H ) or (NLS}.

Similady, we may say the n-vortex is unstable as a solution of one of otu

evolution equations if there exists E > O and initial coriditions arbitrarily close

to the n-vortex such that

for sorne later time. In section 4.5 we preseiit a dynamic instability result

(theorem 6) for the n-vortex with ,\ > 1 and In1 2 2 as a solution of (MH).

This result combines the linearized instability theorern of Chapter 3 with work

of Y. Guo ([G]), and is a weaker statement than the definition of instability

given above.

The dynamical instability of higher-degree (In1 > 2) vortices for (MH) in

the limit of large X was established in [ABG]. Stuart ([SI) studied the (MH)

dynamics of multi-vortex configurations for X close to the critical value ( A = 1).

The basic idea of the stability proof is to use a conserved (or non-increasing)

energy functional as a Lyapunov function, which controls the size of a pertur-

bation. The crucial ingredient in this approach is an estimate on the spectrum

of the Hessian (second variation) of the energy functional around the n-vortex,

E1'(u(")). This estimate, which is proved in Chapter 3: states that under the

conditions of Theorem 5, we have

for any q E HL,, where HL,, is detined to be the space of perturbations

q = (cl a) E H1(R2; C) $ H1(R2; IR2) satisfying the conditions

and

Here

are the zero-eigenvalues of E" ( c l ( * ) ) arising from translat ional symmetry (see

Chapter 3 for more details). These conditions ensure 7 is orthogonal to the

infinite-dimensional kernel Ksym of ER(u(" ) ) due to symmetry breaking.

As we have seen, the presence of (broken) symmetry is a complicating fac-

tor. The Lyapunov technique for proving orbital stability of solutions of non-

linear PDE in the presence of some symmetry has been used often (see [BI ,[We],

[HPW], [GSS] for example). In the present case, the infinite-dimensionality of

the symmetry group contributes a considerable additional difficulty. An argu-

ment based on the implicit function theorem for Banach spaces is used here to

overcome this difficulty.

We remark also that there are a few results available concerning the stronger

property of asymptotic stizbdity for solutions of nonlinear PDEs ([SW], [PW],

[BPI, [WX], [GuP], [BCPS]). In the present situation, such a result is out of

reach for the time being.

The rest of this chapter constitutes a proof of theorem 5, and is organized

as follows. The existence theory for the evolution equations is discussed in

section 4.2. Section 1.3 contains the basic Lyapunov stability argument. The

main technical difficulty in this argument is to show that the solution u ( t ) of

the given equation can always be decomposed as a sum

of a symmetry-modulated n-vortex ( g ( t ) E G,,) and a perturbation which is

orthogonal to the infinite-dimensional kernel of the linearized operator ( ~ ( t ) E

H ) . This is established in section -4.4. Finally, in section 4.5, the question

of instability is dealt with.

4.2 Existence and properties of solutions

We deal first with the Maxwell-Higgs equations (4.8)-(4.10). It is convenient

to work in the temporal gauge (V G O). In this case, the existence result of

[BM] states that for initial conditions of the form

with 70 E H 3 , CO E HZ (here H3 stands for Hy(W2;C) @ H s ( p : Et2)), we have

a unique solution of (MH) of the form

with fj E C(R, HZ), 5 E C(R, Hl) . This solution conserves the energy (Hamil-

t onian) functional

Henceforth, when we refer to a sol~ution of (MH), we mean the one described

here, having assumed initial data of the appropriate form.

In the case of the gradient-flow (1.7) equation, we invoke an existence result

of [DS] . Working again in the temporal gauge, we assume initial data u(t,o =

IL^ = ($O, which is smooth (Cm), and with the additional assumptions

that 1 - 1 + 0 1 2 , mrlAo and Va4, (and their first derivatives) decay with some

exponential rate as 1x1 + m. This guarantees existence of a unique smooth

solution for which the exponential decay of the above-listed field components

is preserved. For this solution, the energy functional is non-increasing:

By a solution of the gradient-flow (GF) equations, we will always mean the

one Sescribed here. Presumably, the exponential decay is not essential to an

existence theorem. It allows, in this case, the proof of a convergence result

using general methods for gradient flows ([DS]).

For (NLS), the author is unaware of an existence theorem. Nonetheless,

we can still formulate a stability theorem. When we refer to a solution of

this equation, we will mean a solution of the form (4.14) with ij E C(R, H2),

6 E C(R, Hl), that conserves the appropriate energy functional

4.3 The stability argument

This section contains the basic stability argument. We use the information on

the Hessian of the energy at the n-vortex furnished by (4.13) to show that the

energi controls perturbations as long as they are orthogonal to the symmetry

zero-modes.

We begin by making a simple adaptation of the basic spectral result (4.13)

for our present purposes.

Proposition 12 Under the conditions of theorem 5. for q E H k t h , we have

ProoE For any O < e < 1, we have

by (4.13). Now E " ( d n ) ) is of the form diag{- A,,,, -A} + V , V a bounded

multiplication operator (see Chapter 3). Thus.

making use of the easy inequality I(VAcni 2 f ( 1 ~ ( ( 1 - ([el(;. Combining this

with (4.17) for sufnciently small E, we obtain the desired result (4.16).

We now establish the key estimate.

Lemma 2 Suppose either

Then there are c, d: e , f: g > O such that if rj E HLth, then

Proof: This result follows easily Erom the full Taylor expansion of E around

dn) (which terminates at fourth order), the estimate (4.16), and the Sobolev

inequali t y

f 0 r 2 ~ ~ < c r , i n ~ * .

It follows irom this lemma and the behaviour of the energy functionals

that small perturbations remain small under the evol~t ion~ provided they stay

orthogonal to the symmetry zero-modes.

Lernrna 3 Given E > O , there is a 5 > O such that if u ( t ) is a solution of

(MH), (GF), o r (NLS) on the time interval [O, Tl, of the f o m

ProoE Let n(t ) = /Iq(t) ( 1 a continuous function on [O, Tl. For concreteness,

consider (MH) (in the temporal gauge, V E O as always). The proof is similar

for (GF) and (NLS). By Lemma 2, and conserved energy (4.15), we have

cn2(t) - dn3 ( t ) - en4(t) 5 H ( u ( t ) ) - ~ ( ~ ( t ) u ( " ) ) (4.20)

There exists > O such that if 1.h.s (4.20) 5 then either O 5 n ( t ) 5 e or

nit) 2 Z, some E > E. By choosing 6 sufficiently small, we ensure both that

r.h.5 (1.21) 5 JO and n(0) 5 E . The result now follows from continuity of n( t ) .

O

In order to prove theorem 5, then, it remains only to prove that a de-

composition of the form (4.19) is possible for al1 time. This is the resuit of

proposition 13 in the next section.

4.4 Imposition of ort hogonality condit ions

This section contains the most technical portion of the stability proof. Our goal

is to show we can mite a solution of the evolution equation in the form (4.19) in

such a way that the modulation of the parameters describing the path g ( t ) d " )

on the manifold G,,dn) is sufficient to ensure rl E H;,,. The tool is the

implicit funct ion t heorem for Banach spaces.

Remark 5 Geometrically, the decomposition (4.19) with the orthogonality con-

dition (q E HLth) means that we are choosing, at each tzme t , the point g ( t )u (n )

on the manifold G,,u(") which 2s L2-closest tu u ( t ) . In fu t , one ,way to estab-

lzsh such a decomposition zs to directly minimite Il u ( t ) - gu(n) 1 1 over g E Ga,, .

The Euler-Lagrange equations for such a minimiration are exactly the desired

orthogonality conditions.

Proposition 13 Let É > O . There zs a b > O svch that a soht ion u(t) of

(MH), (GF), or (NLS) with initial condition uo = IL(") + ~0 with llqo 11 < d

(also IIPollz, llrollz < 6 for (AH) and llPol12 < 6 f o ~ (NLS)) can be written in

the fonn (4.1 9) with q ( t ) E HL,,,, and

for al1 t .

Proof: We adopt the following ansatz for our solution

- i b . , , (") UJ - (Y, +6)

where x : R2 + R, b E and the subscript b denotes a coordinate shift by b

(ie? gb(x) = g ( x - b ) ) . This ansatz represents u = (@, A) as a gauge-transformed

n-vortex plus a perturbation (Jbi ab). The reason for the somewhat complicated

gauge transformation (involving y + b . rlj;") rather then just t ) iç technical. It

is related to the fact that VA(., O(") E L" but Vd") # L2.

We realize the space

H2(R2; IR) x R2. We set

G,, of symmetry transformations explicitly as G =

also G = L?(R2: R) x IR?? and define a map

via

where g = ( x , b ) , 7( t1 g ) = ( [ ( C 9)?a( t , d), and ~ h e r e

m g ) = ,-i[x-,,+b-A(n;]

@ - b ( t ) - dn)

and

By construction. q( t , g ) lies in HL,, (ie satisfies the required orthogonality

conditions) if H ( t g) = 0.

We would like to use the Banach space implicit function theorern (see, eg,

[Be]) to conclude that if H(to, g ( t o ) ) = O for some to, we can solve H ( t , g ( t ) ) = O

for g ( t ) , for t in some neighbourhood of ta. To do this, we need to know that

ag H(to , g( to ) ) is invertible. The following lemma deals with this question.

Lemma 4 H 2s a C L map iuith variational derivative

given by

,wh ere

-A + ( f b ) 2 - ~ ( n f Z b . s')a Lb- (

O m(1d)

is invertible and

Here m > O is a constant, and we recall that f = f,, is the n-vortex profile

function (4.6).

ProoE The (somewhat messy) derivat ive comput ation is relegated to the

appendix (section 4.6), where (4.26) and (4.27) are estabüshed. We can invert

Lb by the operator

- which is bounded as lim,,, f(js1) = 1. LI.

Flow we proceed as Eollows. CVe assume q-, E HL,, (otherwise for 6 suffi-

ciently small, the inverse function theorem (using lemma 4) Eurnishes go such

that u(0) = gou(n) + g with E HLLh and small). So H(0,O) = O . B y

lemma 4 there iç a 8 > O such that if 6 5 & the implicit function theorem

furnishes a CL function g ( t ) on an interval [O, to] with H ( t , g ( t ) ) = O. Using

the continuity of g ( t ) , the HI-continuity of q ( t . g ) in t and g (see section 4.6),

and lemma 3, we can ensure Ilq(t, g ( t ) ) IlHi 5 min(e, 8) by choosing b < 60

(some b0). Now extend the time interwl to the maximal interval [O, T) on

which the C1 function g( t ) satisfying H( t . g ( t ) ) = O exists. If T = oo, then we

are finished, by the above reaçoning extended to [O. m). Suppose, t hen. that

T < oo. Note that atg = -(d,H)-'&H. From (1.26), (-4.27), and the bound-

edness of Ilq(t, g(t))llH1, we have that Il(b,H(t,g(t)))-'11 is bounded uniformly

in t on [O, T) . We also have Il & H ( t , g ( t ) ) 11 bounded uniformly (see section 4.6).

Thus Ilg(t)llc is bounded on [O,T), and so g ( t j ) t g' weakiy in G for some

t j + T. We thus have that H ( T , g * ) = O. From (4.24)-(4.23) it foilows that

q(t,, g(t , ) ) + q(T,g*) weakly in H L , and hence llq(T, g8)1IHl < rnin(e, 6). We

may therefore apply the implicit function theorem to H at (S,g ' ) to obtain

a C L function 3(t) sa t iskng H(t , i j ( t ) ) = O on [T - s , T + s ] , some s > 0.

This 3 constitutes an extension of g ( t ) beyond T, contradicting T < oo. This

completes the proof of Proposition 13, and hence alço the proof of Theorem 5.

O

4.5 A dynamic instability result

Instability is due to negative eigenvalues of EI t (dn) ) which exist when X > 1

and In1 2 2 (Theorem 3 in Chapter 3). That this implies a form of dynamical

instability for the Maxwell-Higgs equations was proven by Y. Guo ( [Gi). We

simply apply this result directly, to obtain the following t heorem.

Theorem 6 When X > 1 and In1 2 2. there is E > O and initial conddions for

any 6 suficiently small satisfying

but for tuhich the solution of satisfies

Slip Il~(t) - ~ ( " ' l l ~ 2 E. t<_CI log61

Here the riorrn is II 1 1 % = II . llii +- II /1&. W e refer the reader to [G] for

the proof, and note only t hat the initial conditions here are ultZo = dn) + 6v,

itZo = 6wv where u is the ground state of &"(IL(")) with (negative) eigenvalue

2 -W .

4.6 Appendk: variational derivat ive calcula-

We collect in this appendix some of the computations and estimates of varia-

tional derivatives of the map H , which are used in section 4.4.

It is a straightforward matter to check that ~ ( t , g ) = ({(t,g), a ( t , g ) ) given

by (4.24)-(4.25) lies in H' for g E G, and that 17 is a continuous map Erom

R x G to H l . The only non-trivial part is a (stilI straightforward) verification

that

and

decay sufficiently quickly to lie in L2. This follows from the form of the n-

vortex (4.6) and the fact that E H2 4 x E L2 n Lm.

Given these facts, it is immediate that H is a well-defined and continuous

map.

When we substitute the form (4.24)-(4.25) of ~ ( t , g) into the definition (4.23)

of the map H we obtain (after some simplification) that the first component

of H ( t , g) is

and the second component is

Using this as our starting point it is a straightforward (if tedious) computation

(involving a few integrations by parts) to obtain

where

wit h

a positive constant. and where the components of R(t. g ) are given by

-&(na'/r)êi ~ ( t . g)).

One can easily check from t hese expressions that Dg H (t, g) : G + G is a

bounded operator, which varies continuously wit h t and g.

It is also more-or-less immediate that

Rom the expression (1.28), it is clear that 11 L;' : G -+ G(( is b~unded inde-

pendently of 6.

Consider, now, & H ( t , g ) . Weobservefirst that li(t,g)l = 1 4 - b l and lQ(t ,g) l =

IAc(. Thus we have

and

whose L' norm we can bound by /IV --.'i(t)(12 + ~l$( t ) I l2. It follows from the

existence theorems for (MH) and (GF) (and our assumptions for (XLS)) that

.-@) is H L-continuous in t' and h(t) iç ~%ontinuous in t . We conclude that

& H ( t , g) is continuous in t and g (recall q ( t , g ) is continuous in t and g), and

t hat Il Ot H ( t . g ) 11 is bounded on finite t ime-intervals independently of g.

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