Physica 26D (1987) 379-384 North-Holland, Amsterdam

INELASTIC SCATrERING OF TODA SOLITONS BY AN INTERFACE

Carlos CAMACHO and Fernando LUND Departamento de Fisica, Facultad de Ciencias Fisicas y Matemdticas, Universidad de Chile, Casil/a 487/3, Santiago, Chile

Received 15 January 1986"

We consider an infinite Toda lattice in which the spring constant on one half differs from the spring constant on the other half. The equations are solved numerically for initial conditions representing a soliton travelling from the left. The resulting (inelastic) scattering by the interface is studied for a variety of incident energies and spring stiffness contrast. We find a markedly different behaviour for incidence on a softer or harder lattice. In the latter case there is efficient soliton generation, with very little energy going into radiation.

1. Introduction

A serious challenge to the physics and mathe- matics of solitons is to understand how these nonlinear objects interact with an external source, forcing, or inhomogeneity. From a mathematical point of view, one is interested in solving a differ- ential equation such as KdV, sine-Gordon, Toda, and the like, with additional terms representing dissipation or external forcing, or in which the coefficients are space dependent. Recent develop- ments along this line include the study of the changes suffered by a solitary wave travelling over a slowly changing topography [1], the interaction of solitons with impurities [2-3] and among each other in non-integrable situations [4-5], and soft- ton behaviour under steady forcing at one end of a semi-infinite Toda lattice [6-7]. From a physical point of view, one has localized lumps carrying energy and momentum and the question arises, how can they be excited and in which way will they evolve in a realistic (which usually means non-integrable) situation. Examples include the excitation of solitons in polyacetylene [8], as well as soliton behaviour in Josephson junctions [9]

*The publication of this article has been delayed consider- ably because of the loss of the manuscript in the mail.

and in stimulated Raman scattering [10]. Also, it would be interesting to understand the fate of elastic solitons [11-12] when the properties of the elastic medium are discontinuous, as at an inter- face. Of course, the difficulty lies in the nonlinear- ity of the problem; linear tools such as Fourier analysis and Green's functions just are not avail- able.

One possible line of attack is to start from the inverse scattering method giving the exact solution to the ideal (i.e. integrable) case and suitably perturb it to obtain information useful in the nonideal case [13]. Another possibility, likely to be useful when deviations from integrability are strong (but where it still makes sense to speak of solitons), is to simple-mindedly solve numerically the appropriate equations and see what happens, in the spirit of a numerical experiment [14]. In this paper we take the second approach; specifically, we are interested in what happens to a soliton as it hits an interface, a surface where the properties of the medium have a discontinuous jump or, more precisely, they change on a scale much smaller than the length of the soliton. This would be the case for an elastic softton incident on the interface between two different media. As a model system for this work we choose an infinite Toda lattice [15] with two different spring constants. Its Ham-

0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

380 C. Camacho and F, Lund/ lnelastic scattering of Toda solitons by an interface

iltonian is

H = 2--- ~ p,

1

+a 2 { b - l ( e - b ( q " + l - q " ) - l ) + q , + l - q , }

+ a ~ o { ( b ' ) - l ( e - b ' ( q " + ' - q " ) - l ) + q n + l - q n

(1)

The energy and momentum of this system are conserved but, to the best of our knowledge, there are no higher conserved quantities. The reason for this choice is that it is already discrete (in space), as appropriate for numerical work, and the case b = b' is integrable and has well known soliton solutions [15]. In our case, b ¢ b', solutions local- ized far to the left or to the fight behave as if the lattice were infinite, and it makes sense to speak of solitons in these regions. A soliton incident from the left will evolve, for large times, into a solu- tion consisting of radiation and new solitons travel- ling to the left ("reflected") or to the right (" t ransmit ted" or "generated by the interface"). We have thus an inelastic scattering of a Toda soliton by an interface and we shall investigate the final states appearing for a variety of initial condi- tions and interface discontinuities. The rest of this paper is as follows: in section 2 we describe the method and give the numerical results. These are analyzed in section 3, and section 4 has final comments and conclusions.

2.0-q(n) 16

12

0.8

' 0.0 - so o s'o ~;o

t

_~_L ~o ~oo 2~o

n

Fig. 1. Solutions of the equations of motion for m = a = 1, (b ,b ' ) =(2 ,30) , time s tep=0.008, after 3000 iterations. The interface is at n = 0. There is one large amplitude reflected soliton travelling to the left, five smaller amplitude transmitted solitons travelling to the right and the slower radiation is in between.

mitted, were generated by the interface, and the evolution was followed until it appeared that soft- ton generation was complete. As a check on the numerics we verified that the energy (1) was pre- served in time to within 0.2%.

Toda solitons centered at the origin at time t = 0 have only one free parameter, say K, and we shall speak indistinctly of their energy, amplitude, or speed. Their explicit form is [15]

mfl 2 e -br" --1 = ~ sech 2 (Kn 4- f i t ) , (2)

where r . = q . - q . _ 1 and f l = ( a b / m ) 1/2 shK. They have an amplitude

q ~ - q+~ = 2 K / b , (3)

which is the property we use to measure K for the scattered solitons. Their energy E and momentum P are

2. Method and results

The dynamical equations for the Hamiltonian (1) were solved numerically by straightforward replacement of time derivatives by finite differ- ences, with initial conditions corresponding to a soliton incident from the far left, and its evolution was followed. A typical situation is shown in fig. 1. In general, new solitons, reflected and trans-

E = ( 2 a / b ) ( s h K c h K - K ) ,

P = -T- 2 ( rna /b ) l / 2 sh K. (4)

A first type of experiment consisted in changing the energy of the incident sofiton with fixed b, b' and a second, changing b' for fixed b and incident energy. The notation K i , K r , K t will refer to the incident, reflected, and transmitted solitons, re- spectively.

C Camacho and F. Lund/Inelastic scattering of Toda solitons by an interface 381

2.1. Incident sofitons of variable energy; spring constants fixed

Fig. 2 shows results for a soliton incident on a softer lattice, b = 2, b' = 0.5, for several K i. There is one reflected and one transmitted soliton and their K ' s are plotted as functions of K i. Results for a soliton incident on a harder lattice are shown in figs. 3 - 5 in order of increasing spring constant contrast.

L

• Kr

0 0 0 0

01 , , ; '. 1.5 20 2.5 30 35

o O

Ki

Fig. 2. K r and Kt are funct ions of K i for a sol i ton inc ident on a sof ter la t t ice: b = 2, b' = 0.5. There is one reflected and one t r an smi t t ed so l i ton in all cases.

Kr

1O

8

6

4

2

o? 2

(a )

~ s 6 7 8 I0

KI

10 •

Kt • o O

• O O

0 [ - - I I I I I I 2 3 4 5 6 7 8 9 I0

Ki

Fig. 4. The case b = 2, b' = 5 at vary ing K i. As in the case of fig. 3 there is one reflected (a) and two t ransmi t ted (19) soli tons,

w i th the l ead ing sol i ton looking like the inc ident one wi th a

more severe d is tor t ion .

Kr 2-° 1 (a) L5

tO

05

oo¶ 2 3 /, 5 6 7 8

Ki

';I10 ' Kr

2~ •

ol 2 B 9 ,0

Ki

~0 (b) 8

5

Kt t. •

2q

__9 3 L

O o

Ki

Fig. 3. The case b = 4, b ' = 4.5 at vary ing K i. There is one ref lected so l i ton (a) and two t ransmi t ted ones (b). The s imilar- i ty of b - b ' is reflected in the fact that the leading t r ansmi t t ed so l i ton c losely resembles the inc ident one.

12

10

8

K t 6 L

2

(b) O

[] [] []

t - - 0

Ki

Fig. 5. The case b = 2, b ' = 20 at vary ing K i. There is one ref lected so l i ton (a) and four (resp. three) t ransmi t ted ones (b)

w h e n K i _< 6 (resp. K i ~ 6).

382 C. Carnacho and F. Lund/ lnelastic scattering of Toda solitons by an interface

2.2. Incident sofiton of fixed energy; b fixed, b' variable

Results for K i -- 2, b = 2 are shown on fig. 6, in which K r and K t are plotted in terms of b'. There is one reflected soliton and the number of trans- mitted solitons increases with b' from one to five. Fig. 7 has similar information for g i ---6. In this case we did not find a reflected soliton. The fea- tures of the transmitted solitons are similar to the

case K i = 2.

Kr

(a) 6

L

2

4 6

• • • • • •

, , , i L

8 ,o 12 ,6 ,'8

b'

3 . D i s c u s s i o n

For the parameter range considered in this work, soliton behaviour at an interface appears to be markedly different according to whether it hits a

harder ( b ' > b) or softer ( b ' < b) lattice. For a

3° tzo (o)

Kr

J lO [O ° O o °

0 0 ~ _ _ , 0 2 5

• • •

t - - - - I

b'

30,

20 J

Kt -

lO-

OOt 0 2

(b)

0 0 0 • • • • O 0 e e o o

g O 0 • D 0 •

• []

0

5 ~0 I~

0 0

[] []

[] •

2C] 25 30 b'

Fig. 6. One inc iden t sol i ton of K i = 2, b = 2 wi th b' variable. (a): K r in t e rms of b', there is only one reflected soli ton; (b): there are f rom one to five t ransmi t ted sol i tons whose K t are p lo t t ed in te rms of b'.

6-

4

K t , 2

0

oo • , ° " ° 8 8 8 i ~ 0 0

0 0 • •

0 •

o • • {b)

0 •

• [ ]

- - o ~ []- i 5 10 15 20 25

b'

Fig. 7. Fo r an inc ident sol i ton of K i = 6, b = 2, var iable b' there is one reflected sol i ton (a) if b' > 2 and from one to four t r an smi t t ed ones (b).

soliton incident on a softer lattice, it is seen from fig. 2 that there is only one reflected and one transmitted soliton, whose energies increase with

the energy of the incident soliton. K r increases more slowly than K t in accord with the intuition that an energetic soliton should not be very dis- turbed by the interface. For incidence on a harder lattice the number of transmitted solitons in- creases, being higher the harder the lattice. How- ever, for b ' = 20 we found that the number of transmitted solitons decreased as the incident en- ergy increased. This, and the fact that the energy of trailing transmitted solitons increases with K i more slowly than the energy of the leading trans- mitted soliton (cf. figs. 3-5) confirms the intuition already mentioned that soliton of high enough energy should hardly be bothered by an interface. A quantitative assessment of this intuition is ob- tained by noting that at fixed b, b' the leading soliton tends (as K i increases) to a situation de-

C. Camacho and F. Lund/ Inelastic scattering of Toda solitons by an interface 383

sc r ibed by

K t ( l ead ing) = (1.0002 + 0 .0004)K i

- (0.6940 + 0.0004),

b = 2 , b ' = 0 . 5 ,

K t ( l e a d i n g ) = (1.0001 +__ 0 .0004)K i

+ (0.058 _+ 0.003),

b = 4 , b ' = 4.5,

g t ( l ead ing) = (1.0007 + 0.0008) g i

+ (0.449 + 0.006),

b = 2 , b ' = 5 ,

so tha t for h igh K i the leading t ransmi t t ed sol i ton

does indeed take most of the energy, with the

in te r face hav ing a small, and independen t of K i,

effect. In the case b = 2, b ' = 2 0 we could not

push our c o m p u t a t i o n to high enough K i to reach

an a s y m p t o t i c regime.

A n o t h e r po in t wor thy of not ice and in which

so l i ton behav iou r for incidence on a softer or

h a r d e r la t t ice is qual i ta t ively different, is the

a m o u n t of energy that is scat tered in the form of

r a d i a t i o n (i.e. not soliton). The percentage of inci-

den t energy so scat tered is ind ica ted in table I.

F o r b ' < b and low K i a significant f ract ion of the

inc iden t energy is rad ia ted away, and it rap id ly

decreases as K i is increased: again, the interface is

on ly a smal l pe r tu rba t ion for a h igh-energy soli-

ton. W h e n b ' > b the energy that goes into radia-

t ion is negligible.

A t fixed inc iden t energy, figs. 6 - 7 show that the

n u m b e r of t r ansmi t t ed soli tons s teadi ly increases

wi th b ' for b ' > b. That is, a s teep interface is an

efficient sol i ton generator . F o r K i = 2, b ' < 2 there

is on ly one reflected soli ton whose energy in-

creases as b ' decreases, that is as the interface gets

sharper , bu t there is also a lot of rad ia t ion (see

tab le II). F o r g i = 6, b ' < 2 we d id not find a

ref lected sol i ton, in all l ikel ihood because it would

t ake more t ime than we have avai lable for it to

Table I Percentage ( _+ 0.2%) of incident energy scattered as radiation for b, b' fixed and variable K i

b = 2 b = 4 b = 2 b = 2 K i b ' = 0 , 5 K i b' = 4, 5 b' = 5 b' = 20

2 22.3 2 0.3 1.3 1.6 3 5.6 4 <0.2 0.6 1.1 4 1.2 6 <0.2 <0.2 1.2 5 0.2 8 <0.2 0.2 1.6

Table II Percentage ( + 0.2%) of incident energy scattered as radiation for fixed b, K i and variable b'. A star ( * ) indicates that this quantity could not be measured.

b' K i = 2 K i = 6

0.5 22.3 * 1.0 7.4 * 4.0 1.4 0.3 6.0 1.4 <0.2 8.0 1.6 0.6

10.0 1.6 0.3 12.0 1.6 1.1 20.0 1.7 1.2

384 C, Camacho and F. Lund/ lnelastic scattering of Toda solitons by an interface

appear. Again, for b'> b the incident soliton is scattered mostly into new solitons, the energy going into radiation being less than 2% (table II).

4. Concluding remarks

We have computed the inelastic scattering of Toda solitons by an interface separating two semi-infinite Toda lattices of different spring con- stants. Apart from the detailed quantitative be- haviour described above, the main qualitative observation is that, for incidence on a harder lattice, most of the incident energy is scattered as solitons with a very small amount of radiation, the number of solitons generated at the interface in- creasing as the difference in spring constants in- creases. In other words, a sharp interface is a very efficient soliton generator. It should be interesting to see whether this qualitative and quantitative behaviour is peculiar to the Toda lattice or. it is shared by other soliton bearing systems. A par- ticularly appealing candidate for study is the sine-Gordon chain.

Acknowledgements

This work was supported by DIB Grant E 1963-8522, Fondo Nacional de Ciencias Grant 1147-1984 and UNDP-UNESCO Program CHI- 84-005.

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