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INTERACTION OF MULTICOMPONENT SOLITONS IN ATOMIC CHAINS
S. Cadet
To cite this version:
S. Cadet. INTERACTION OF MULTICOMPONENT SOLITONS IN ATOMIC CHAINS. Journal
de Physique Colloques, 1989, 50 (C3), pp.C3-21-C3-28. �10.1051/jphyscol:1989303�. �jpa-00229442�
JOURNAL DE PHYSIQUE
Colloque C3, supplement au n03, Tome 50, mars 1989
INTERACTION OF MULTICOMPONENT SOLITONS I N ATOMIC CHAINS
Laboratoire d'optique du RBseau Cristallin, Faculte des Sciences, 6 bd Gabriel, F-21100 Dijon, France
RQsumQ
-
Nous nous interessons aux mouvements purement transversaux d'une chaine atomique. Les solutions des equations dynamiques, obtenues dans une approximation semi-discrhte, ont la forme de modes enveloppes polarisr5s circulairement et rectilignement. L'Btude numerique de leur stabilite montre que les enveloppes polarisees circulairement se comportent comme des solitons alors que les modes polarises rectilignement subissent une variation de polarisation pendanc leur interaction. Les resultats numkriques sont interpretes analytiquement ?l'aide d'une imethode perturbative.
Abstract
-
We investigate the purely transverse motion of an atomic chain. Solutions of the dynamical equations that are obtained in a semidiscrete approximation, have the form of circularly polarized and linearly polarized envelope modes. Numerical treatment of their stability show that circularly polarized envelopes have soliton- like behaviour while linearly polarized modes suffer variations of polarization during mutual interaction. These results are studied analytically with the help of a perturbative method.1
-
Introduction.The construction of stable structures is a major issue in nonlinear science. Of particular importance are nonlinear dispersive partial differential equations that admit soliton solutions. These infinitely long lived waves are renowned for their remarkable stability, in particular two solitons interact elastically: they preserve their asymptotic identity through mutual interaction without emission of radiation. Several nonlinear equations of interest have been extensively studied by means of the inverse scattering transform and Hirota's method. These two methods make it possible to obtain multisoliton solutions that represent the whole interaction process.
Unfortunately most physical systems are modeled by more complex evolution equations which reduce to integrable equations only provided some appropriate conditions are fulfilled. The soliton solutions then yield approximate solutions of the original problem and the stability has to be verified numerically or analytically when possLble. It is found that in most cases the approximate solutions adapt quickly to the syrtem while loosing a small amount of energy in the form of linear oscillations. Moreover the solixary waves thus obtained exhibit almost soliton-like behaviour: negligible radiation is emitted as a result of their collision and each wave suffer only asymptotic phase shifts. Eut it may also happen that the approximation fails to capture some essential featurs o P the model which results in some instability.
In the course of an investigation of the three-dinensional dynamics of' an atomic chain we were confronted with such a situation / I / . The transverse nonlinear motion is governed by a difference-differential equation coupled with the longitudinal motion that satisfies a linear equation forced by transverse relative displacements. In order to obtain nonlinear modes of the chain we employed a standard procedure /2/ that consists in looking for solutions in the form of an oscillation modulated by an envelope function. Clearly.
owing to the cylindrical symmetry of the problem, a transverse oscillation can be chosen circularly polarized (CP) or linearly polarized (LP) in any direction. In both cases the envelope function satisfies a NonLinear Schrijdinger equation (NLS) which has soliton solutions /3,4/. A numerical study of collisions between envelope modes then shows two kinds
"'Present address: Laboratoire de Modelisation en Mdcanique, Universite P. et M. Curie, tour 66, 4, place Jussieu, F75252 Paris Cedex 05, France
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989303
C3-22 JOURNAL DE PHYSIQUE
of d e v i a t i o n from t h e behaviour of s o l i t o n s . F i r s t , a resonance due t o frequency mixing may occur between t r a n s v e r s e and l o n g i t u d i n a l o s c i l l a t i o n s , r e s u l t i n g i n an important emission of l o n g i t u d i n a l r a d i a t i o n . Besides, envelopes i n i t i a l l y p o l a r i z e d l i n e a r l y emerge from c o l l i s i o n with an e l l i p t i c p o l a r i z a t i o n . The l a t t e r phenomenon i s examined i n t h i s paper where w e r e s t r i c t t o p u r e l y t r a n s v e r s e motion which i s more e a s i l y - d e a l t with.
While t h e i n v e r s e s c a t t e r i n g transform and H i r o t a ' s method a r e adequate when d i s c u s s i n g t h e c o l l i s i o n of waves t h a t a r e s o l u t i o n s of t h e same e q u a t i o n , they can not be applied here. Indeed, w e c o n s i d e r s o l i t a r y waves t h a t s a t i s f y independently d i f f e r e n t NLS e q u a t i o n s , t h a t i s with d i f f e r e n t c o e f f i c i e n t s of d i s p e r s i o n and n o n l i n e a r i t y , because they may have d i f f e r e n t wavevectors o r propagate i n o p p o s i t e d i r e c t i o n s o r have d i f f e r e n t p o l a r i z a t i o n s . I n o r d e r t o work o u t t h e a n a l y s i s of i n t e r a c t i o n s , we u s e ' a r e f i n e d version of t h e r e d u c t i v e p e r t u r b a t i o n method /5,6/ t h a t we extend t o t h e d i s c r e t e problem i n t h e cases of CP-CP and LP-LP i n t e r a c t i o n s . Owing t o t h e l e n g t h y a l g e b r a involved, we p r e s e n t only a s k e t c h of t h e c a l c u l a t i o n s t h a t can be found i n more d e t a i l s elsewhere /7/.
2
-
Model e q u a t i o n s and s o l i t a r y waves.We c o n s i d e r an i n f i n i t e chain of p a r t i c l e s of mass m i n t e r a c t i n g with t h e i r f i r s t n e a r e s t neighbours and l y i n g a t r e s t along a l i n e with t h e l a t t i c e s p a c i n g a . Here we r e s t r i c t t o t h e c a s e where each mass can move o n l y i n t r a n s v e r s e d i r e c t i o n s and we denote by (Y,, Z,.) t h e two components of t h e displacement of t h e n t h mass from i t s equilibrium p o s i t i o n ( F i g u r e 1 ) .
Figure 1: P i c t u r e of t h e chain d e f i n i n g t h e v a r i a b l e s . The Hamiltonian of t h e system is
where dn i s t h e bond l e n g t h between neighbours n and ( n + l ) :
The p o t e n t i a l is chosen i n t h e form
V(d) = V, (d-a) + V, (d-a),
where t h e f i r s t term i s due t o t h e permanent i n t e r n a l s t r e s s V, and t h e second term s t a n d s f o r t h e harmonic p a r t o f t h e i n t e r a c t i o n . The c o e f f i c i e n t s V, and V2 a r e p o s i t i v e i n o r d e r t o e n s u r e t h e s t a b i l i t y of t h e e q u i l i b r i u m s t a t e a g a i n s t p e r t u r b a t i o n s . I t i s n o t necessary t o i n c l u d e m a t e r i a l n o n l i n e a r i t y i n t h e p o t e n t i a l s i n c e a g e o m e t r i c a l n o n l i n e a r i t y i s p r e s e n t owing t o t h e n o n l i n e a r dependence of t h e bond l e n g t h on t h e d i s p l a c e m e n t s a s s t a t e d by ( 2 . 2 ) .
The e q u a t i o n s o f motion t h a t d e r i v e from t h e Hamiltoniqn ( 2 . 1 ) cannot b e solved a n a l y t i c a l l y . We o b t a i n more t r a c t a b l e e q u a t i o n s when t h e r e l a t i v e d i s p l a c e m e n t s a r e small a s compared t o t h e l a t t i c e spacing: we can then expand dn with r e s p e c t t o t h e d i f f e r e n c e s Yn
+,
-Yn and Zn+, -Zn SO t h a twhere we n e g l e c t h i g h e r o r d e r terms because t h e second power is s u f f i c i e n t i n view of t h e f o l l o w i n g c a l c u l a t i o n s .
It i s convenient t o i n t r o d u c e t h e complex v a r i a b l e Hn which d e s c r i b e s both components o f motion: Hn = Yn + i Z n
.
The approximate e q u a t i o n s o f motion t h e n read:
where Cg = V, a/m , Cf = V 2 a 2 / m
,
(2.6.a.b)t h e c o e f f i c i e n t s C, and C, a r e t h e speed o f sound f o r t r a n s v e r s e and l o n g i t u d i n a l l i n e a r waves r e s p e c t i v e l y and we f u r t h e r m o r e assume C,
<
C,.F i g u r e 2: P l o t o f CP and LP envelopes.
C 3-2 4 JOURNAL
DE
PHYSIQUEWe s t i l l cannot produce an e x a c t s o l u t i o n of t h e d i s c r e t e s e t of d i f f e r e n t i a l equations (2.5) but we can o b t a i n approximate a n a l y t i c s o l u t i o n s of i n t e r e s t .
I n p a r t i c u l a r we can look f o r s o l u t i o n s i n t h e form of a slowly varying envelope function F modulating a small harmonic wave with frequency w and wavenumber k r e l a t e d by t h e l i n e a r d i s p e r s i o n r e l a t i o n w = w ( k ) . The problem is e a s i l y d e a l t with when t h e harmonic wave i s e i t h e r c i r c u l a r l y p o l a r i z e d o r l i n e a r l y p o l a r i z e d i n any d i r e c t i o n . I n both c a s e s , going t o t h e continuum l i m i t and i n t r o d u c i n g s u i t a b l e v a r i a b l e s , i t is found t h a t t h e envelope function F i s governed by a NonLinear Schrodinger equation
where P.Q
>
0 and t h e c o e f f i c i e n t s P and Q depend on t h e parameters of t h e c h a i n , the wavenumber k and t h e p o l a r i z a t i o n of t h e wave.Going back t o t h e o r i g i n a l v a r i a b l e s , t h e s o l u t i o n / 8 / of t h e NLS equation y i e l d s displacements i n t h e form of a c i r c u l a r l y o r l i n e a r l y p o l a r i z e d o s c i l l a t i o n t h e amplitude of which has t h e shape of a pulse ( F i g u r e 2 ) .
However, while t h e s o l u t i o n s of (2.7) a r e s o l i t o n s , i t i s n o t a p r i o r i c e r t a i n t h a t t h e a n a l y t i c expressions deduced f o r t h e d i s c r e t e system s h a r e t h e same s t a b i l i t y p r o p e r t i e s . Indeed numerical simulations show t h a t head-on c o l l i s i o n s of two LP waves r e s u l t i n t h e i r p o l a r i z a t i o n becoming e l l i p t i c whenever t h e i n i t i a l d i r e c t i o n s of l i n e a r p o l a r i z a t i o n make an a n g l e d i f f e r e n t from 0 o r ~ / 2 (Figure
3 ) .
S i m i l a r numerical r e s u l t s a r e obtained by Parker /9/ f o r s i g n a l s propagating i n o p t i c a l f i b r e s .On t h e o t h e r hand, c o l l i s i o n s between CP and LP waves o r between two CP waves (Figure 4) show no such behaviour. Both envelope modes only undergo phase s h i f t s .
Figure 3: P l o t of t h e t r a n s v e r s e displacement: ( a ) before and ( b ) a f t e r a head-on c o l l i s i o n between two LP s o l i t a r y waves of same amplitude and d i r e c t i o n s of p o l a r i z a t i o n making an angle of 45".
3- I n t e r a c t i o n of two CP envelopes.
We assume t h a t Hn i s t h e superposition of two c i r c u l a r l y polarized envelopes and we d i s c a r d t h e harmonics r e s u l t i n g from t h e combination of t h e two waves a s t h e i r c o n t r i b u t i o n w i l l n o t appear i n t h e following equations:
Figure
4:
Plot of the transverse displacement after a collision between two CP solitary waves of the same amplitude (b). In (a) a single wave which has travelled unimpeded.where a, = ~ ( k , ) satisfy the linear dispersion relation and &
<<
1 is a smallness parameter.We furthermore assume the group velocities vi=
a~
-(k=ki) to be such that Iv,-
v21 is of the akorder of unity.
Substituting from (3.1) in (2.5) and going to the continuum limit we get:
-(l-cos8,) (1-cos8,) IF, I2Fl+ ai.sinel (1-cos8,) IF, I2Fl,
el e 2 el +82 13.21
-
2ai. sin-. sin-. sin-2 2 2 F, F;. xF, + 2ai. sin-. 2 sin-. 2 s i n 7 F; F, , F,
c; - c;
where g =
-
and Bi = kia.2a4
Next we introduce the stretched coordinates
and the form of Fi is anticipated as:
f:O) + ~ f i l ) + &,fi2) +
. . .) ex. i(.~:" + ~'0:" + ...)
C3-26 JOURNAL DE PHYSIQUE
where f: j ) , Jr:j ) and 0: J ) a r e f u n c t i o n s of El
,
E, and r ; Jr$ j ) and R$ j ) a r e supposed t o be r e a l .The v a r i a b l e s 0:j) and R $ j ) a r e introduced i n e x p e c t a t i o n t h a t t h e v e l o c i t i e s of the two wave packets and t h e i r frequencies vary i n space and time because of t h e i r nonlinear i n t e r a c t i o n .
Equating t h e c o e f f i c i e n t s of t h e v a r i o u s powers of & i n t h e same harmonics t o zero we g e t t h e sequence of equations t o be solved, which we g i v e only f o r t h e wave with frequency ol a s t h o s e obtained f o r t h e o t h e r wave with frequency o2 a r e s i m i l a r .
The f i r s t equation a t o r d e r & i s s a t i s f i e d owing t o t h e d i s p e r s i o n r e l a t i o n . order E' :
t h i s i m p l i e s t h a t t h e l e a d i n g term i n t h e expansion ( 3 . 3 ) i s not a f f e c t e d by the i n t e r a c t i o n ; i t s dependence on El and r i s s p e c i f i e d a t t h e next s t e p .
~ r d e r 8 : we rearrange t h e v a r i o u s terms i n t h r e e equations.
*from a n o n s e c u l a r i t y c o n d i t i o n we recover t h e NLS equation f o r a s i n g l e wave, which determines f i O )
.
af; 1 )
* - - -
0 which s t a t e s t h a t t h e i n t e r a c t i o n does n o t a f f e c t f i l ) * the Second term in the 352expansion of t h e envelope.
*
a r e l a t i o n determining t h e v a r i a t i o n of t h e phase Ri') due t o mutual i n t e r a c t i o n :where g, i s a c o e f f i c i e n t t h a t depends on t h e v e l o c i t i e s C,, C, and t h e wavenumbers of t h e i n t e r a c t i n g waves.
order & 4 i s rearranged i n four equations t h a t give:
*
the dependence of f I 1 ) on El and r ;*
t h e v a r i a t i o n o f t h e phase a t second o r d e r R i 2 ) ;*
t h e s h i f t i n t h e p o s i t i o n due t o t h e i n t e r a c t i o n :*
the v a r i a t i o n of amplitude due t o t h e i n t e r a c t i o n :where u, depends on t h e parameters of t h e chain and t h e wavenumbers.
The important p o i n t t o n o t e is t h a t t h e dependence of f i 2 ) on E2 is p r o p o r t i o n a l t o a product of t h e amplitudes a t f i r s t o r d e r of t h e two envelopes. Thus t h e v a r i a t i o n of
amplitude i s l o c a l i z e d i n t h e r e g i o n where t h e two waves o v e r l a p and i t vanishes a s y m p t o t i c a l l y when t h e y a r e s e p a r a t e d b e f o r e and a f t e r t h e c o l l i s i o n .
We could proceed t o n e x t o r d e r s a t t h e p r i c e of more and more i n t r i c a t e a l g e b r a i c c a l c u l a t i o n s . However t h e computation up t o t h i s o r d e r is s u f f i c i e n t t o e x h i b i t t h e l e a d i n g terms of phase and p o s i t i o n s h i f t s , a s d e r i v e d from ( 3 . 7 ) and ( 3 . 8 ) . which a r e found t o be i n agreement w i t h numerical v a l u e s o b t a i n e d from computer s i m u l a t i o n s .
4-
Interaction of two LP envelopes.I n t h i s c a s e i t is n o t s u f f i c i e n t t o look f o r s o l u t i o n s Hn i n t h e form of t h e s u p e r p o s i t i o n o f two LP waves s i n c e we wish t o d i s c u s s a v a r i a t i o n of p o l a r i z a t i o n . Accordingly we c o n s i d e r t h a t Hn i s t h e sum of two waves with any p o l a r i z a t i o n , each wave being s p l i t i n a l e f t h a n d e d F i , and a righthanded
e i ,
c i r c u l a r l y p o l a r i z e d component:where a is a r e a l number t h a t w i l l r e p r e s e n t t h e a n g l e between t h e d i r e c t i o n s of p o l a r i z a t i o n i n t h e s p e c i a l c a s e of l i n e a r p o l a r i z a t i o n where
F i e ,
= F:. n.The p r o c e s s used i n t h e p r e c e d i n g s e c t i o n can be r e p e a t e d , t h a t i s , i n s e r t i n g ( 4 . 1 ) i n t h e d i f f e r e n t i a l e q u a t i o n (2.5). g o i n g t o t h e continuum l i m i t w h i l e r e t a i n i n g only t h e terms r e l e v a n t t o t h e f o l l o w i n g c a l c u l a t i o n :
Then i n t r o d u c i n g t h e new c o o r d i n a t e s ( 3 . 2 ) . expanding t h e amplitude and t h e phase of each envelope f u n c t i o n ( 3 . 3 ) . f i n a l l y e q u a t i n g t h e c o e f f i c i e n t s o f t h e v a r i o u s powers of E i n t h e same harmonics t o z e r o l e a d s t o t h e e q u a t i o n s t o be s o l v e d .
The sequence o f e q u a t i o n s t h u s d e r i v e d i s r a t h e r s i m i l a r t o t h a t presented i n t h e c a s e o f CP waves. A t o r d e r E~ i t i s found again t h a t t h e l e a d i n g term i n t h e expansion of t h e amplitude of e a c h envelope i s n o t a f f e c t e d by t h e c o l l i s i o n :
A s a consequence, i n view of s t u d y i n g t h e i n t e r a c t i o n o f waves t h a t i n i t i a l l y a r e l i n e a r l y p o l a r i z e d , i t i s assumed t h a t :
A t o r d e r E 3
,
f i O ) i s determined a s a s o l u t i o n of a NLS e q u a t i o n and t h e v a r i a t i o n of t h e phase i s o b t a i n e d . B u t , c o n t r a r y t o t h e c a s e of CP envelopes, t h e r e i s a v a r i a t i o n of amplitude a t t h i s o r d e r . From a n o n s e c u l a r i t y c o n d i t i o n we o b t a i n a f t e r an i n t e g r a t i o n :C3-28 JOURNAL DE PHYSIQUE
Relations ( 4 . 5 ) show t h a t t h e lefthanded and t h e righthanded c i r c u l a r l y p o l a r i z e d components of a wave undergo o p p o s i t e v a r i a t i o n of amplitude t h a t do n o t vanish when t h e waves move away: t h i s r e s u l t s i n t h e growth of e l l i p t i c p o l a r i z a t i o n .
5- Conclusion.
We have thus shown t h a t t h e chain i s capable of supporting s o l i t a r y waves i n the form of c i r c u l a r l y and l i n e a r l y p o l a r i z e d envelopes. Our a n a l y s i s of t h e i r i n t e r a c t i o n s i n d i c a t e s t h a t CP waves a r e s t a b l e while LP envelopes should r a t h e r be considered a s bound s t a t e s of two CP components which s u f f e r o p p o s i t e v a r i a t i o n s of amplitude during LP-LP c o l l i s i o n . Moreover t h e values of t h e s h i f t i n p o s i t i o n , r o t a t i o n and v a r i a t i o n of amplitude prove t o be i n good agreement with t h e numerical r e s u l t s . I t would be of t h e utmost i n t e r e s t t o check whether CP envelopes a r e e x a c t s o l i t o n s i n t h e d i s c r e t e l a t t i c e . Unfortunately, t h i s i n v o l v e s very i n t r i c a t e c a l c u l a t i o n s when r e t a i n i n g a l l t h e r e l e v a n t terms: t h e r e is need t o expand H,+, i n a s e r i e s , consider a l l t h e harmonics r e s u l t i n g from t h e mixing of the i n t e r a c t i n g waves, i n t r o d u c e higher s c a l e s i n time and space and f i n a l l y prove a t any order
E P t h a t t h e c o r r e c t i v e terms o t h e r than phase s h i f t s vanish asympotically.
With t h e h e l p o f t h e p e r t u r b a t i v e method we have discussed a simple i n s t a n c e of c o l l i s i o n between envelopes b u t t h e question whether any process of i n t e r a c t i o n can be addressed i s open. However i t is i m p l i c i t l y assumed t h a t t h e r e is no l o s s of energy i n the form of r a d i a t i o n . I t i s l i k e l y t h a t t h e method can be applied t o any nonlinear equation whenever envelope modes a r e considered though t h e c a l c u l a t i o n s may be s t i l l more i n t r i c a t e than i t is t h e c a s e when adding t h e l o n g i t u d i n a l motion t o t h e model (Cadet, unpublished).
On t h e o t h e r hand t h e r e i s another important c l a s s of s o l i t a r y waves, pulses and kinks, t h a t has n o t been examined i n t h i s paper. When p a s s i n g t o t h e continuum l i m i t , we can d e r i v e from
(2.5) a Complex Modified Korteweg-deVries equation:
This equation admits l i n e a r l y p o l a r i z e d p u l s e s o l u t i o n s
/lo/
t h a t a r e n o t s o l i t o n s /11/:a f t e r a c o l l i s i o n t h e i r p o l a r i z a t i o n i s modified and r a d i a t i o n i s produced. Besides Oikawa and Yajima t r e a t e d t h e i n t e r a c t i o n of counter t r a v e l l i n g p u l s e s applying t h e p e r t u r b a t i o n approach /5/ t o a weakly d i s p e r s i v e Klein-Gordon equation. This suggests t h a t a b e t t e r understanding could a l s o be gained concerning t h e v a r i a t i o n of p o l a r i z a t i o n of t h e pulse s o l u t i o n s of ( 5 . 1 ) .
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