M. Eleftheriou, B. Dey and G.P. Tsironis- Compacton Like Breathers in Nonlinear Anharmonic Lattice with Nonlocal Dispersive Interaction

8/3/2019 M. Eleftheriou, B. Dey and G.P. Tsironis- Compacton Like Breathers in Nonlinear Anharmonic Lattice with Nonlocal D

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Available at: http://www.ictp.trieste.it/~pub-off IC/2000/19

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

COMPACTON LIKE BREATHERS IN NONLINEAR ANHARMONICLATTICE WITH NONLOCAL DISPERSIVE INTERACTI ON

M. EleftheriouDepartment of Physics, University of Crete

and Foundation for Research and Technology-Hellas,P.O. Box 2208, 71003 Heraklion, Crete, Greece,

B. DeyDepartment of Physics, University of Crete

and Foundation for Research and Technology-Hellas,

P.O. Box 2208, 71003 Heraklion, Crete, Greece,

Department of Physics, University of Pune, Pune 411007, India 1

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

G.P. TsironisDepartment of Physics, University of Crete

and Foundation for Research and Technology-Hellas,P.O. Box 2208, 71003 Heraklion, Crete, Greece.

Abstract

We show the existence and stability of localised breather modes with compact support in ananharmonic lattice with power dependence r~ s of the dispersive interactions. We also considerthe effect of the harmonic nonlocal dispersive interaction on such solutions.

MIRAMARE - TRIESTE

March 2000


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1 Introduction

It is now well known that asa result of the interplay between discreteness, dispersion and non-linear interaction intrinsic localized modesor discrete breathers are inducedin translational

invariant Hamiltonian lattice. Intense work during the last ten years [?] has addressed and resolved in many cases issues regarding their rigorous existence, numerical constructions, stabilitydynamics, thermodynamics, quantum aspects and very recently also the experimental manifestation in specific materials[?]. One aspect tha t th e discrete breathers seem to share in most caseand with other nonlinear localized excitations like solitons in continuum systems is the typicaspatial exponential profile which gives them an infinite spatial extent. However, recently Rosenau and Hyman introduced [?] the concept of compactification or strict localization of solitarywaves. They discovered that in the continuum systems, the solitary waves can be compactifiedin the presence of nonlinear dispersion and termed these solitary waves with compact supporas compactons. The possibility of existence of such intrinsics localized modes or breathers witcompact support in discrete nonlinear anharmonic lattice was predicted in recent studies [?, ?]More recent work [?] has shown thata class of exact continuous compacton solutions ofa loworder continuous approximationof the discrete equations of motion, survive in general whensubstituted in the corresponding discrete equations. What is remarkable is that,it appears, thecontinuous cosine-shape com pacton solutions seems to represent quite well breathe r solutions ithe highly discrete regimes far from th e continuous limit. In ref. [?]it is shown that the discretecompact breathers in the nonlinear anharmonic lattice can be generated in the anti-continuoulimit through a numerically exact procedure and are generally found to be stable.

In all the above studies of compact breathersin discrete lattice, the non linear dispersiveinteractions are assumed to be short range and thereforea nearest neighbor approximationisused. However, there exist physical systems suchas ionic and molecular crystals where thisapproximation cannot describe appropriately the physical systems. For example, excitationtransfer in molecular crystals and energy transportin biopolymers [?] are due to transitiondipole-dipole interaction with^ dependence on the distancer. Also, the DNA molecules havelong range Coulomb interaction between them as the molecules contain charged groups.Thecompact breathers would be ideal for energy storage since due to the lack of exponential tail theywould not interact until they are in contact with each other, thus increasing their lifetime andensuring practical applications such as energy transport, signal processing and communications

However, in these applications, the long range interactionis important, as, for example, innonlinear fiber optics, the long range interaction imposesa strict limitation on the performanceof long haul fiber transmissions [?].

Some studies of solitons and intrinsics localized mode (conventional discrete breathers withexponential tails) in the presence of long range interaction are known. For example,it hasbeen shown [?] that solitary waves like kinks or gap solitons are effected quantitatively by thelong range interactions. Gaidideiet al [11] also considered the effect of harmonic nonlocal in-teraction potential in a chain with short range anharmonicity and observed the existence oftwo types of solitons with characteristically different width and shape for two velocity region


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breathers (conventional types). For example, a general proof of the existence of breathers ind- dimensional latt ice with algebraic decaying interaction are considered in [?]. Similarly, forth e discrete nonlinear Schroedinger (DNLS) model with long range interaction varying with thedistance as r~ s, it has been shown that [?] for sufficiently large s, all features of the model are

qualititavely the same as the D N L S model with short range (nearest neighbor) interaction, butfor s less than some critical s cr there is an interval of bistability. Similarly, Bonart [?] consid-ered the intrinsic localized modes in the presence of Coulomb interaction. These studies raisedan important question: Can a discrete breather with compact support survive the effects of along range interact ion? In this letter we address this problem. We demonstrate the existence

of discrete breathers with compact support in the anharmonic nonlinear Klein Gordon lattice

with onsite nonlinear substrate potential and with power dependence r~ s on the distance rof the nonlinear dispersive interaction. The stationary state of the systems are studied bothanalytically and numerically. We obtain exact analytic compacton breather solutions of the cor-

responding equations of motion in the continuum limit in the presence of long range interactionan d nonlinear substrate potential. These analytic solutions are then used as an initial conditionfor the numerical solutions of the discrete equations of motion and numerical simulations showt h a t within some parameter range the compacton breather solutions survive in the presenceof lattice discreteness and long range anharmonic interactions . The range of the long rangeinteraction parameter s within which the analytic continuum solutions are shown to exist agreesquite well with that obtained from the numerical calculations.

2 Equ ation s of m otion an d exact a n alytic solution s in th e con-

t inuum limitT h e model we study is described by the Hamiltonian

2m n

s2

n,m

where the potential V( n) is a nonlinear onsite potential, n (t) = n is the displacement ofth e n- th unit mass oscillator from its equilibrium position at time t, k1 and k2 determine the

strength of the harmonic and anharmonic interaction between the oscillators respectively and s1an d s2 are the long range interaction parameters, being introduced to consider different physicalsituations ranging from nearest neighbor (s = oo) , dipole- dipole interaction (s = 3), Coulombinteraction (s = 1) etc. Tuning the parameters k1 and k2 as well as s1 and s2, we can studyth e effect of competition between anharmonicity and dispersion and the interplay of the longrange interaction and lattice discreteness and also between harmonic and anharmonic interaction

between th e particles. The equations of motion are given by

m m n|s1


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I t is a difficult problem to obtain th e analytic solution to these equations of motion. For

th e case of k2 = 0 (i.e. the case of conventional exponentially decaying discrete breathers inth e presence of long range interaction), the analytic solutions have been obtained earlier byapproximate methods like the variational method [?] and lattice Greens' function method [?].

The lattice Green's function method describes only the asymptotic properties of the solutions.Implementing such methods for the present problem with k2 = 0 (i.e in th e presence of long

range nonlinear dispersive interact ion) is a difficult proposition. However, we realized th at if we

are only interested in the properties of the localized solutions with analytic behavior, like th e

exponentially localized solutions, compactons etc., th en such solutions in the continuum limit

can easily be obtained by a simple Taylor series expansion of the discrete equations of motion.Accordingly, we rewrite eq.(2) as

O O / I , I nl \

V

U n + p ~ n? + (< t>n-p~ n]

and expand n p in terms of Taylor series as

^ V ( * )+ f / ' " W+ (4)

To show that the Taylor series expansion as given above can actually give the analytic local-ized solutions, we used this expansion to reproduce the exponentially localized solutions of the

D N L S with harmonic long range interaction which was earlier obtained by Rasmussen et alusing the approximate variational method [?]. For this we substitu te equation (4) above in

th e corresponding discrete equations of motion of D N L S with harmonic long range interaction(eq.(4) in [?]) and then it can easily be checked that the corresponding continuum equations of

motion have localized solutions given by (x) = \/ 2Xsech[ j}?^) ], where (s) is the Riemannzeta function, s is the long range interaction parameter and J is the strength of the long range

harmonic interaction. We have also checked that the energy and the excitation number N of th esystem which corresponds to this localized solution also agrees exactly with the corresponding

values obtained with th e variational m ethod (eq.(18) in [?]). We now show th at the compact

breather solutions of the discrete equations of motion (eq.(2)) in the continuum limit can also

be obtained using the Taylor series expansion as above. For this we substitu te eq.(4) in eq.(3)

above to get the corresponding continuum equations for t he system as ( we first consider the

case of k 1 = 0, i.e. no harmon ic interaction)

- a - / 3# + 3 f eC f e - 4 ) ( f ^ g | (5)

To get the compact breather solution we use the ansatz


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Substituting this in eq.(5) we get the equations for y(x) and G(t) respectively as

^v JK dx'

G + G + CG 3 = 0 (7)

I t can easily be checked that the above equation supports compacton breather solutions given

by

(x,t) = A cos(Bx) cn[(A 2 + a)^^/2t,k 2],

for \Bx\ 5 and 0 < s 2 < 2.

Now we consider th e case when th e harmonic displacive interact ion is present (i.e. k / 0).

I n the absence of long range interaction, it has been shown [?, ?] that the harmonic displacive

interaction destroys th e compactons in the discrete anharm onic lattice by progressively turning

compact brea th ers into convention al expon ent ially decaying ones. However, in the presence

of long range interaction, the situation may be different. To examine th is, we introduced inth e Hamiltonian (eq.(2)) two sets of parameter (k1,s1) and (k 2 , s 2 ), so that by suitably tuning

these parameters we can compare the effects of different con tributions from the harm onic and

anharmonic terms in forming the stable compacton breather solutions. Using the Taylor series

expansion as above, the continuum equations in this case is given by

Using the ansatz (x, t) = y(x)coswt for the compacton breather solutions, the continuoumequations reduces to

A2

33

+ f c [ < ( 2

where 2 = w 2 and we have used the rotating wave approximation to get to this equation.I t can again be checked that this equation has compacton solutions of the form


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where

A

=

( g i- 4)

( s 2 - 4 )J l ;

Thus, from the condition of real A and B we see that for the appropriate choice of parameter

values, compacton breather solutions of the form ( x,t) = Acos(Bx)coswt may exist in the

continuum limit for the system containing both the harmonic and anharmonic long range dis-persive interactions. We would like to point out that the first derivatives of these solutions is

discontinuous at the edge and hence the compacton solutions presented here must be understood

in th e weak sense. The robustness of these compacton solutions are yet unknown. H owever, as

reported by Rosenau (see ref.12 in [?]), extensive numerical studies of th e continuum equat ionsindicate th at compactons smoothness at the edge is not indicative of their stability.

3 Numerical analysis

To see whether the above analytic continuum compacton breather solutions survive in the dis-

crete lattice, we have obtained the numerical solutions of the discrete equations of motion

(eq.(2)). The initial condition for the numerical solutions is chosen as n(t = 0) = Acos(Bn)where B is the inverse width of the continuum compacton breather as given above. The ini-

tial velocity is taken t o be zero. To take in to account th e effect of th e long range interaction

effectively, th e numerical simulations are done over a large latt ice with N = 1000 latt ice sites.

Similarly, to check the stability of th e solutions over time, th e solutions are evolved over a

very long time. F irst we consider th e case when no harmon ic term s are present, i.e. k1 = 0.

Fig.(1) shows th e evolution of th e lattice profile of th e compacton breather solutions over a time

t = 150T, where T is the period of the compacton breather. The initial width of the compactons

are chosen to be Lc = B = 6 times th e lattice spacing, where B is given by eq.(9). The longrange parameter s 2 is chosen to be s 2 = 6, th e amplitude as 0.1 and the results are plotted in

Fig.(1a). As can be seen from these figures, th e initial analytic cont inuum compact breather

solutions (eq.(8)) remains stable even after a very long time ( t= 150T) in th e discrete latt ice.

Fig.(1b) is the same as in Fig.(1a) but with s 2 = 7. Again we see that the compact breathersolutions remain very stable. We have checked t hat the compacton breather solutions for other

values of the long range parameter s 2 > 4 also remains stable. We have also considered the

compact breather solutions with width Lc = 30 times the latt ice spacing. In th is case the re-

sults of the numerical simulations show that although the solutions remain stable initially for a

time t = 50T (Fig.(2a)), it loses its compact support and develops some structures near its edge

after a larger time t = 150T (Fig. (2b)). We also observe th at t he stability of th e compactonbreather solutions in th e discrete lat tice depen ds on the amplitude of the solutions. For exam-

ple, the stable compacton solutions with amplitude 0.1 as shown in Fig.(1a) becomes unstable


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the parameter range s < 4 but with amplitude much smaller than similar solutionsfor s > 4.Fig.(1c) shows the discrete compacton breather solution with amplitudeA = 0.01 for s = 1.This solution becomes unstableif the corresponding amplitude is increasedto 0.1 as in Fig.(1a)and F ig.(1b). Finally, the results of the num erical simulations of the discrete equations of moti

in the presence of both the harmonic and anharmonic long range dispersive interactions (bok1,k2 / 0) again shows that the discrete compacton breather solutions, even though stablefora time t = 50T initially, loses its compact structure after aboutt = 150T . This is shown in Fig.(4) for the parameter valuess1 = s2 and k1 < k2 (Fig.(4a)) and s1 > s2 and k1 < k2 (Fig.(4b)).

4 Conclusions

In conclusion, we have examined here the questionof existence and stability of the compactonlike breather solutions in the nonlinear anharmonic lattice with nonlocal dispersive interactiowith power dependencer~ s on the distance. Using a simple Taylor series expansion we derivthe equations of motion in the continuum limit and show that the resulting equations suppocompacton breather solutionsin the presence of the nonlocal dispersive interaction. We thennumerically solve the discrete equationsof motion and observe that the existence and stabilityof the compacton like breathersin the anharmonic lattice with long range interaction dependcrucially on the value of the long range interaction parameters as well as on the amplitude andwidth of the solutions. In absence of the harmonic dispersive interaction, the stable compactolike discrete breathers existfor certain ranges of the long range interaction parameter(s > 4).For s < 4, the simulations on the discrete lattice show that the compacton like breather solutions are stable but with much smaller amplitude than thatfor s > 4. This might indicate that

there may be other kinds of solutions (non compact type) for this parameter range. This is alsuggested from the resultsof the earlier studies on harmonic lattice (DNLS) with long rangeinteractions of the typer~ s [?, ?], that, there isa transition from the exponential decaying solutions to algebraic decaying solutions around the parameter values = 3. Finally, we consideredthe effect of the nonlocal harmonic interaction on the stable compactons as obtained above.Wefind that, even though the solutions are stable initially,it starts developing a tail near the edgeof the compacton, thereby destroying the compact natureof the solutions. This is similar tothe results obtained from the earlier studiesof discrete compacton breathers without the lonrange interaction [?, ?] which showed that the presenceof the harmonic dispersive interaction

progressively turns compact breathersto conventional exponentially breathers.

AcknowledgmentsOne of the authors (B.D.) would like to thank the Physics Department, Universityof Crete,

Greece, for their kind hospitality during his visit, where partof this work was done. Partia lsupport from INTAS 96-0158 as wellas support from Universityof Crete is also acknowledged


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Figure 1: Plot of temporal behaviour of spatial profile of a compacton breather of width Lc6 , (a) s2 = 6, A=0.1 (b) s2 = 7, A=0.1 (c) s2 = 1, A=0.01. The central site is at N=500. The

solutions are stable.

Figure 2: Same as in Fig.(1a) but with Lc = 30; temporal evolutions after (a) t=50T and (t=150T. The initial spatial profile of this large width compacton breather loses its shape neits edges.


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Figure 4: Plot of the tem poral evolution of spatial profile of a compacton breathe r in the presenceof both harmonic and anharmonic long range interactions. Here Lc = 6, N=1000 and t=150Tfor (a) s1 = s2 and k1 < k2 (b) s1 > s2 and k1 < k2. The initial compacton solutions loses itsshape after a long time.


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(b)

1


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l i iu l l fff rn l l l l l ' I f

485


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1

(b)

x 150T

Documents

M. Eleftheriou, B. Dey and G.P. Tsironis- Compacton Like Breathers in Nonlinear Anharmonic Lattice with Nonlocal Dispersive Interaction