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Oscillations of a highly discrete breather with a critical regime

E. Coquet, M. Remoissenet, and P. Tchofo DindaLaboratoire de Physique de lUniversite de Bourgogne, UMR CNRS No. 5027, Avenue A. Savary,

Bote Postale 47 870, 21078 Dijon Cedex, France

Received 24 January 2000; revised manuscript received 30 June 2000

We analyze carefully the essential features of the dynamics of a stationary discrete breather in the ultimate

degree of energy localization in a nonlinear Klein-Gordon lattice with an on-site double-well potential. Wedemonstrate the existence of three different regimes of oscillatory motion in the breather dynamics, which are

closely related to the motion of the central particle in an effective potential having two nondegenerate wells. In

given parameter regions, we observe an untrapped regime, in which the central particle executes large-

amplitude oscillations from one to the other side of the potential barrier. In other parameter regions, we find the

trapped regime, in which the central particle oscillates in one of the two wells of the effective potential.

Between these two regimes we find a critical regime in which the central particle undergoes several temporary

trappings within an untrapped regime. Importantly, our study reveals that in the presence of purely anharmonic

coupling forces, the breather compactifies, i.e., the energy becomes abruptly localized within the breather.

PACS numbers: 41.20.Jb, 05.50.q

I. INTRODUCTION

In recent years there has been considerable effort made in

understanding nonlinear energy-localization phenomena 1

in homogeneous discrete lattices. The possibility of existence

of localized long-lived molecular vibrational states waspointed out by Ovchinnikov 2 many decades ago. Then, theexistence of nonlinear localized modes in a lattice with on-site potential and linear interparticle coupling was exploredby Kosevich and Kovalev 3. Dolgov 4 proposed a modelfor self-localization of vibrations in a one-dimensional latticewith nonlinear interparticle coupling without on-site poten-tial. Sievers and Takeno 5 reported that large amplitudevibrations in perfectly periodic one-dimensional 1D lattices

can localize because of the nonlinearity and discreteness ef-fects. Then they clearly suggested that intrinsic localizedmodes or discrete breathers should be quite general and ro-bust solutions in the sense that they can exist in many models5. This result was subsequently confirmed by Page 6. Incontrast to the case of purely harmonic lattices, where spa-tially localized modes can occur only when defects or disor-der are present so that the translational invariance of theunderlying lattice is broken, discrete breathers may be cre-ated anywhere in a perfect homogeneous nonlinear lattice.Since the pioneering studies mentioned above 1 6, discretebreathers have been the subject of an intense research forrecent reviews see 7 and 8, because of their relevance in

various fields such as condensed matter physics, optics, andbiology. The existence of discrete breathers as exact timeperiodic spatially localized solutions in a large class of non-linear lattices was demonstrated analytically by McKay andAubry 9. But so far, no analytical derivation of breathersolutions in a highly discrete -four lattice has been carriedout because of a host of mathematical problems that arise inthe specific situation of abrupt localization of energy.

On the other hand, recent studies 10,11 demonstrate thata breathing mode can compactify in a nonlinear Klein-Gordon lattice, with a soft on-site substrate potential single-well potential, in the presence of purely anharmonic cou-

pling forces between adjacent particles. The concept of

compactification or strict localization of solitary waves wasintroduced for the first time by Rosenau and Hyman 12,

who investigated a special type of Kortewegde Vries equa-

tion and discovered that solitary waves can compactify in the

presence of a nonlinear dispersion. Such solitary waves,which are characterized by a compact support, i.e., the ab-sence of infinite wings, have been called compactons. Then afundamental question arises as to whether a breather com-pacton can exist in nonlinear Klein-Gordon lattices havingan on-site double-well substrate potential. The answer to thisquestion is given in the present paper.

In this paper, we carry out a theoretical analysis showingthe existence of a stationary breather in the ultimate degree

of energy localization in a nonlinear Klein-Gordon discretelattice, with a -four on-site substrate potential and veryweak coupling forces between adjacent particles. Thisbreather consists of a central particle that executes stronglyanharmonic dynamics while all the other particles executevery small-amplitude oscillations about their equilibrium po-sitions. Such a situation, which presents similarities with thelocalized rotating modes 13, may occur in many discretereal systems where some thermally activated atoms or ions,though coupled to their neighbors, may jump from one site toanother one. In the present paper, we provide a general pic-ture describing the oscillatory dynamics of the highly dis-crete breather, and we show that the on-site double-well po-

tential leads to a much richer spectrum of behavior than thereis for the single-well substrate potentials considered in pre-vious studies 10,14. Most of the richness comes from theexistence of several regimes of breather motion, which areclosely related to the motion of the central particle of thebreather in a double-well effective potential. Many physicalprocesses involving this dynamics may include proton trans-port in hydrogen-bonded chains 15, planar rotations of basepairs in DNA macromolecules 16, and polymer chainstwisting 17.

The paper is organized as follows. In Sec. II we presentthe model. In Sec. III, we analyze the breather motion in the

PHYSICAL REVIEW E OCTOBER 2000VOLUME 62, NUMBER 4

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untrapped regime, and in Sec. IV we consider the trappedregime. Section V is devoted to the analysis of the criticalregime, and in Sec. VI we summarize and give concludingremarks.

II. MODEL

A. Equations of motion

The system under consideration is a nonlinear Kein-Gordon infinite lattice, in which each particle interacts withits nearest neighbors via coupling forces that are eitherpurely harmonic or anharmonic. Each particle oscillates in adirection perpendicular to the chain axis, in a -four on-sitepotential that represents the combined influence of the sur-rounding lattice atoms and external effects. The latticeHamiltonian is

Hn

12

dxndt

2

c l

2xn1xn

2

c nl

4 xn1x n4

V0

4 1xn2

2

. 1Here, xn is the on-site degree of freedom, c l , c nl , and V0

are constants that control the strength of linear and nonlinearcouplings, and the on-site potential barrier height, respec-tively. The corresponding one-dimensional equations of mo-tion can be written in the following form:

d2x n

dt2V0xn 1x n

2Clx n1x n12x n

Cnl xn1x n3x n1xn

3 , 2

where Clc l /V0 and Cnl c nl /V0 are the normalized cou-pling constants.

In the present paper, we examine the ultimate degree ofenergy localization in this nonlinear Klein-Gordon lattice, inthe form of a breather. To this end, we consider the limit ofweak coupling between adjacent lattice sites i.e., a dynami-cal situation in which only a central particle CP, say ns, executes a large-amplitude motion from one to the otherside of the potential barrier. All the other particles executerelatively small-amplitude oscillations near the bottom of thesubstrate wells lying in the same side of the potential barrier.Figure 1 shows a schematic representation of the configura-

tion of the system at the beginning of this dynamical situa-tion. The CP is shifted to an unstable position in one side ofthe substrate potential, labeled B in Fig. 1. As a result ofthe coupling forces, the particles neighboring the CP undergosmall shifts, in the same direction as the CP, up to unstableequilibrium positions located near the bottom of the potentialwells labeled A in Fig. 1. An important point to be em-phasized here is that the large initial displacement given tothe CP, in the direction perpendicular to the chain axis, gen-erates thereby a symmetry plane perpendicular to the chainaxis, at this lattice site i.e., s. Consequently, in the limit ofno perturbations, the amplitude of displacement of particlessi and si are strictly identical. Thus, at site ns, onecan replace Eq. 2 by

d2x s

dt2V0x s 1xs

22 Clxs1x s Cnl x s1xs

3.

3

On the other hand, the positions of all the particles, exceptthe CP, can be rewritten as: xsi1xsi , where thedeviation with respect to the equilibrium position isx

s

i1. Then, for C

l

1 and Cnl

1, the equations ofmotion at sites s, s1, and s2 reduce, in first-order ap-proximation, to

d2x s

dt2V0x s 1xs

2 2 Cl 1x s Cnl 1x s3

dVex s

dx s, 4

d2xs1

dt22V0xs1V0Cl 1xs Cnl 1x s

3 ,

5

d2x s2

dt22V0xs20. 6

As Eq. 6 shows, in the weak coupling limit, the motion ofthe s2 particles corresponds essentially to an harmonicoscillation with frequency 0(2V0)

1/2 and period T02/0 .

In Eq. 4, Ve represents the effective potential for the CP,whose expression is given by

Vex V0 14 1x2 2Cl 1x 2Cnl

2 1x 4 , 7

FIG. 1. Schematic representation of the initial configuration of

the system in presence of a weak interparticle coupling. The curves

represent the on-site substrate potential. The circles represent theparticles. The labels A and B identify the two wells of the

potential. The central particle, which lies at the site index s,

begins its motion in the well B, whereas all the other particles are in

the wells A. The particle position x is given in arbitrary units.

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with xx s . Thus, Eqs. 46 define the motion of the par-ticles that are involved in the breather dynamics in the ulti-mate degree of energy localization in the system. The effec-tive potential Ve is schematically represented by the solidcurve in Fig. 2, where the positions of the extrema of Ve :x em 1 , x em 2 , and x em 3 , can be easily calculated by solvingdVe(x)/dx0 the results are given in Table I. One of theeffects of the coupling forces is to lower the barrier height of

the potential VbhVe(x em 2)Ve(xem 3) that the CP mustovercome to move from the well B to the well A. In fact, theability of the CP to execute a large-amplitude motion de-pends critically on its initial position x0 . There exists a criti-

cal distance x 0ec from the origin from which the CP when

released with zero initial velocity can execute a large-amplitude motion from one to the other side of the potentialbarrier. Note that in the absence of coupling forces, the ef-fective potential of the CP coincides with the on-site sub-strate potential represented by the dotted curve in Fig. 2, and

there, the critical distance is x 0c&x 0

ec . Thus, an out-standing effect of the coupling forces is the reduction of the

critical distance from x0c to x 0

ec see also Table I. Throughout

the present paper, we choose the initial position of the CP tobe precisely in the parameter region x0x 0

c , where the pres-ence of coupling forces is required for the CP to execute anuntrapped motion. In this region, the breather motion can bebroadly divided into three main regimes of oscillatory mo-tion depending on the initial position of the CP.

i If x 0x0ec , the CP will execute an untrapped motion

with large-amplitude oscillations.

ii When x 0x0ec , we find a trapped regime in which the

potential energy of the CP is not sufficient for this particle toovercome the potential barrier.

iii When x0x 0ec , we find a critical regime in which

different situations can occur depending on the precise value

ofx 0 , such as the occurrence of several temporary trappingswithin an untrapped regime.

We examine successively these three different regimes.

III. UNTRAPPED REGIME

In this regime, the CP has a sufficient energy to overcomethe potential barrier. Thus, Eq. 4 can be integrated via aquadrature to give

x w et,ke2 x0

ppcn w et,ke2

cn w et,ke2

, 8

where the parameters w e , ke2, p, , and , are given in Table

II for zero, linear and nonlinear coupling forces, respectively.The function cn in Eq. 8 is a Jacobi elliptic function 18

with modulus ke21. Solution 8 corresponds to oscillations

of period Te and frequency :

Te4

w eK ke

2 with K ke

20

/2 du

1ke2 sinu

2and

2

Te. 9

For very weak coupling: Cl1 and Cnl1 including thecase when ClCnl0, the general solution 8 can be ap-proximated 18 by

x tx 0cn w et,ke x0 n0

P 2n1 cos 2n1 t,

10

where

P2n1

keK ke2

cosh1 n 12 K 1ke

2

K ke2

. 11

Solution 10 shows that the central-particle motion,which is closely related to the breather motion, is made up ofthe fundamental frequency and several odd harmonics.This behavior is formally equivalent to large-amplitude os-cillations of a particle in a double-well potential 19,20. Thesubstitution of the expressions for w e and ke see Table II inEq. 9 gives the period Te in terms of the initial condition x 0

FIG. 2. Schematic representation of the effective potentials for

the central particle Ve solid curve, and for the breather Veffdashed curve. The dotted curve represent the substrate potential.

The parameters x em 1 , x em 2 , and x em 3 , indicate the extrema of the

effective potential of the central particle. x1ec and x0

ec indicate the

positions for which Ve(x)Ve(x em 2) .

TABLE I. Parameters of the effective potential of the CP Ve(x),

for zero, linear and nonlinear coupling cases. The positions of the

extrema of Ve(x) are: x em 1 , xem 2 , and x em 3 . Vbh is the barrier

height. x0ec is the critical initial position of the CP.

Parameters

No

coupling

Linear

coupling

Nonlinear

Coupling

x em 1 1 1 1

xem 2 0 2Cl 2Cnlxem 3 1 12Cl 18CnlVbh V0

4

V0

4112Cl

V0

4 130Cnl

x0ec

& & 1Cl(1&) & 1Cnl (43 &)

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and the coupling parameters. The dotted-dashed curve in Fig.3 represents the normalized period Te /T0 for oscillations of

the CP as a function of its initial position x0 , in the absenceof coupling. Te /T0 becomes infinite when the initial position

of the CP coincides with the critical value: x0x 0c& as

indicated in Fig. 3 by a vertical dotted line. Thus, in theabsence of coupling, the CP would be unable to cross the

potential barrier for initial positions such that x 0x0c . In the

presence of coupling, Te is represented by the solid curves inFig. 3, which show that the CP is now able to cross the

potential barrier for initial positions below x 0c , in the region

x 0ec x 0x0

c . Here, the coupling plays a crucial role in the

sense that it is required for the CP to execute large-amplitudeoscillations from one to the other side of the potential barrier.On the other hand, the achievement of the ultimate degree of

energy localization in the breather imposes a strict limitationin the strength of linear or nonlinear coupling, to ensuresmall-amplitude oscillations for all the particles other thanthe CP. As a consequence, the size of the parameter region

x0ec x 0x0c must be sufficiently small. Hereafter, we useCl6.50010

4 and Cnl1.909104, for which x 0

ec

1.411 99.Although the CP plays the prominent role in the motion of

a highly localized breather mode, the role of the other par-ticles of the system cannot be ignored in a discrete system,because they are involved in the energy-exchange or com-pactification processes, which we discuss below. An analyti-cal description of the motion of these particles ( s1) maybe obtained by rewriting Eq. 5 as follows:

d2xs1

dt20

2x s1V0G t, 12

where the function G in the right-hand side of Eq. 12 isgiven by

G tG l tG nl t

Cl 1x tCnl 1x t3

Cl 1x 0 n0

q

P 2n1 cos 2n1 tCnl 1x 0

n0

q

P 2n1 cos 2n1 t3

, 13

TABLE II. Characteristic parameters w e , ke2, p, 2, and 2, for

the CP motion in the untrapped regime, for zero ( Cl ,Cnl )(0,0),

linear (Cl,0) and nonlinear (0,Cnl ) coupling.

(Cl ,Cnl ) we(Cl ,Cnl )

0,0 V0(x021)

(Cl,0)we0,0

1

Cl

x02

12 x0

22x01

(0,Cnl )we0,0 1 Cnl

x0212

[x042x0

3x0

2x03]

(Cl ,Cnl ) ke(Cl ,Cnl )2

0,0 x02

2x021

(Cl,0)ke0,0

2 1 2Clx0x0

21 2

x03x02

(0,Cnl )ke0,0

2 1 4Cnlx0x0

21 2

2x03x0

22x01

(Cl ,Cnl ) p(Cl ,Cnl )

0,0 1

(Cl,0)1 4Cl

x0x021

(0,Cnl )

1 4Cnl1x02

x0x021

(Cl ,Cnl ) (Cl ,Cnl )

2

0,0 2(x021)

(Cl,0) 2(x021)4Cl

(0,Cnl ) 2(x02

1)

8Cnl (2

x0)

(Cl ,Cnl ) (Cl ,Cnl )2

0,0 2(x021)

(Cl,0)2x0

214Cl1 4x0

x021

(0,Cnl )

2x0218Cnl2x0x0

23

x021

FIG. 3. Plot showing the normalized period of oscillations

Te /T0 in the effective potential Ve , as a function of the initial

position of the central particle x0 , for Cl6.5104 and Cnl0

solid curves, T0 being the period of small-amplitude oscillations

in the bottom of one the two degenerate wells of the substrate

potential. The dotted-dashed curves represent the period of oscilla-

tions in the on-site substrate potential Cl0 and Cnl0.


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q is an integer whose value depends on the order of approxi-mation that is desired. The functions G l( t) and Gnl (t) can be

rewritten into the general form

G Q ti0

J

g iQ cos it, 14

where the index Q stands for l or nl. The coefficients g iQ aregiven in Table III for the highest-amplitude harmonics (i0,1,2,3,4). Keeping only these harmonics, with the initialconditions: xs1(0) 0, and xs1(0) 0, we obtain thefollowing approximate solution for Eq. 12:

xs1 t n0

J

Xn sin 1 nm 0t

2sin nt

2 , 15

where we have set

m0

, n 1 nm 0 , Xn

m2

m2n 2gnQ .

16

Equation 15 shows that the motion of the particles ( s1) depends strongly on the coupling parameters ( Cl ,Cnl )through Xn , which depends on gnQ . Furthermore, the am-plitudes Xn in Eq. 15 become infinite when m0 / is anonzero integer. This resonant process is only qualitativelycorrect because of the approximate nature of Eq. 15. A

resonant phenomenon can thus occur in the system if ahigher-order harmonic of the breather coincides with the fre-quency of small-amplitude oscillations in the bottom of oneof the two wells of the substrate potential, 0 . All of theabove qualitative considerations are remarkably well-confirmed by the numerical simulations that we have per-formed using the equations of motion of the system, Eq. 2.The numerical method used to obtained the initial configu-ration of the breather, as schematically represented in Fig. 1,is the so-called relaxation process 21. We accomplishthis relaxation by, first, putting the CP to a given position x 0in the potential well labeled B in Fig. 1, and all the otherparticles in the bottom of the wells A. Then, we let all theparticles except the CP to move according to Eq. 2. Peri-

odically, the velocities of all those particles are set to zeroafter a specified number of time steps. In doing so, we peri-odically extract energy from the system until all the particlescease to move. Once this process is completed, the effectivepotential Veff of the breather can be obtained from the fol-lowing expression taking into account all particles of thechain:

Veffn

V0 Cl2 x n1xn 2 Cnl4 x n1xn 4

1

4 1x n

22 , 17

which is schematically represented by the dashed curve inFigs. 2, in a very qualitative way. Indeed, this potential dif-fers clearly from the effective potential Ve of the CP solidcurve in Fig. 2 only when the strength of the coupling forcesis sufficiently important. In the weak coupling limit underconsideration throughout the present paper, Veff is essentiallyidentical to Ve . Hereafter, unless otherwise specified, we

take V01.

A. Linear coupling

Figures 4, that we have obtained for x 01.414, show that

as soon as the CP is released from a position x0x 0ec

1.411 99, it executes a large-amplitude oscillatory motionfrom one to the other side of the potential barrier see Fig.4a1. In Figs. 4a1, 4b1, 4c1, and 4d1, we have rep-resented the motion of the CP s and its neighbors: s1, s2, s3, and in Figs. 4a2, 4b2, 4c2, and 4d2, theFourier spectra of each motion, respectively. These spectrareveal that the breather motion is made up of a fundamental

frequency and a few higher-order harmonics n. But,only odd harmonics contribute to the breather motion andthose harmonics are present in the motion of all the particlesthat make up the breather, i.e., the CP and its nearest neigh-bors s1. Furthermore, we see that particles s2 and s3 oscillate at the phonon frequency 0 with amplitudesthat increase in time at least in the first stage of the dynam-ics. This behavior indicates that a small amount of the en-ergy that was initially localized on the three particles, s ands1, is progressively transferred to other lattice sitesthrough a phonon-radiation process. Indeed, owing to thelinear coupling, a phonon band: k0 12Cl sin

2 k/21/2, 0k, exists in the frequency spec-trum of the system, whether the breather is present or not inthe lattice. In the very weak coupling limit that we consider,the upper phonon band edge lies very close to 0 . Thus, thebreather motion acts as a source of excitation of the phononspectrum, leading to small oscillations at frequency 0 forparticles that lie away from the breather, as Figs. 4c2 and4d2 show. The conversion of the CP energy into phonon-mode energy occurs in a relatively smooth and continuousmanner low-conversion regime.

As mentioned above, the breather motion involves a fun-damental frequency that always lies in the gap below thelower-phonon band edge: 0 . Consequently, as de-pends on the initial position x0 of the CP see Fig. 3, ahigher-order harmonic of may for some specific values of

TABLE III. Characteristic parameters g iQ for linear Ql and

nonlinear Qnl coupling, for the motion of particles s1 in the

untrapped regime, described by Eqs. 14, 15, and 16.

Linear

coupling (g il ) Nonlinear coupling (g in l)

g0lCl g0nlCnl

1

3x02

2

P12P3

2

g 1lClx0P1 g1nl3Cnlx0P11x0

2

4P1

2P1P32P3

2g2l0 g2nlCnl

3x02

2P1

22P1P3

g 3lClx0P3 g3nlCnlx0P13

43x0P31x0

2

4 P3

22 P1

2 g4l0 g4nl3Cnlx0

2P1P3


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x 0 make direct resonance with phonon modes, thus produc-ing a strong transfer of energy away from the breatherstrong-conversion regime. Such a situation is representedin Figs. 5, where for x 01.4122 we have represented themotions of the CP and particles s1 and their respectiveFourier spectra. Note that the size of the phonon band is sosmall that only a single harmonic can be present inside this

phonon band 0n024Cl

1/2. The resonancephenomenon observed in Fig. 5a2 results from the presenceof the harmonic n5 in the phonon band, which causes alarge enhancement of the amplitude of oscillations for allparticles except the CP, in particular for particles that arefar away from the CP. For example, in the strong-conversionregime, the amplitude of oscillations for the particles s2and s3 is by more than one order of magnitude larger thanin the low-conversion regime compare Fig. 4c1 with Fig.5c1, or Fig. 4d1 with Fig. 5d1. However, it is worthnoting in Fig. 5a1 that this strong conversion regime doesnot cause a substantial drop in the amplitude of the CP os-

cillations, owing to the weakness of the coupling forces be-

tween adjacent particles.

B. Nonlinear coupling

For x 01.414, Cl0, and Cnl1.909104, Figs. 6a1

and 6a2 show that, the CP motion exhibits the same generalfeatures as for the linear coupling cases considered aboveFigs. 4a1 and 4a2, and Figs. 5a1 and 5a2 in the sensethat only odd harmonics of the breather motion are present inthe CP motion. Figs. 6b1 and 6b2 reveal the presence ofall harmonics odd and even in the motion of the nearestneighbors of the CP, whereas only odd harmonics are presentin the linear coupling case see Figs. 4b1 and 4b2, and

Figs. 5b1

and 5

b2

. Moreover, we see in Figs. 6

a1

,6b1, and 6c1 that the amplitude of oscillations of particles

lying away from the CP decreases abruptly from xs1 forthe CP to x s210

10, and no appreciable motion is de-tected for particles s3. This abrupt localization of energywithin the breather demonstrates the ability of a breather to

compactify in a nonlinear Klein-Gordon lattice with an

on-site double-well potential. However, we are unable toprove analytically that we have a compactonic behaviorin a strict mathematical sense. Nevertheless, such breatherswill be referred to as compactons in a physical sense i.e., adiscrete breather with abrupt localization. In this context,the energy localization on only three particles ( s,s1) cor-responds to the ultimate degree of energy localization in a

FIG. 4. Plots showing the temporal evolution of the positions in

arb. units of particles that make up the breather motion, obtained

by solving exactly the discrete equations of motion 2, for x01.414, Cl6.510

4 and Cnl0. a1: xs(t). b1: x s1( t)

1xs1( t) . c1: xs2(t)1x s2(t). d1: x s3(t)1x s3(t). t designates the time in arb. units. a2, b2, c2, and

d2 show the Fourier transforms of the motion represented in a1,

b1, c1, and d1, respectively. Each peak in a2 and b2 corre-

sponds to an harmonic n of the breather motion. In a2, the

labels 1, 3, 5, and 7 indicate the harmonics that are

present in the CP motion.

FIG. 5. Plots obtained in the same conditions as for Figs. 4, but

with x01.4122.


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discrete breather. Furthermore, Fig. 6b2 shows that thechosen initial particle position x01.414 corresponds to asituation where the fourth harmonic of the breather motionresonates with phonon modes. As a result, the amplitudes ofoscillations of particles s2 see Fig. 6c1 are found to be,

by more than one order of magnitude, larger than the corre-sponding amplitudes in any off-resonant case in the regionx 01.414. Thus, the compactification of the breather is pre-served even in the presence of the resonance phenomenon, aslong as the coupling forces are sufficiently weak.

As a general remark for the whole untrapped regime, it isworth noting that the deviation between the analytical ex-pression of the CP motion in Eq. 8, x(t), and the numericalsolutions Xs(t), measured by nXs(n dt)x(n d t) 2/x 0 where dt is the time step does not exceed2%. Equation 8 therefore provides a highly accurate repre-sentation of the CP motion in the untrapped regime.

IV. TRAPPED REGIME

In this regime, the CP starts out at a position x 0x0ec ,

with a potential energy that is below the critical level forcrossing the barrier of the effective potential: Ve(x0)Ve(xem 2) see Fig. 2. The CP is therefore trapped in theshallower well while all the other particles oscillate in thedeepest well. This situation, which has been largely ne-glected in all previous discretized field theories of a breathermotion, makes one of the greatest qualitative differenceswith respect to the breather motion in a lattice with a single-well on-site potential in which only a single regime exists10. In the weak coupling limit, Eq. 4 can be integrated viaa quadrature, in a similar way as for the untrapped regime in

Sec. III. The solution, in terms of the Jacobi elliptic function

dn 18 with modulus ke21, may be written as

x tx0dn wet,ke2 , 18

where we(Cl ,Cnl )we(0,0)x 0(V0/2)1/2. The parameters

ke2(Cl ,Cnl ) are displayed in Table IV. Solution 18 corre-

sponds to oscillations of period Te and frequency :

Te2

we

K ke2 with K ke

2

0

/2 du

1ke2 sin2 uand

2

Te

. 19

In the weak coupling limit, Eq. 18 can be rewritten as seeRef. 18:

x tx 0 2K k2

n1

Q2n cos 2n t , 2/Te ,20

where

Q2n

K k2cosh1 n F 1k2

K k2 . 21

On the other hand, we have found from numerical simu-lations that in the trapped regime the breather exhibits thesame general features in both linear and nonlinear couplingcases, except that in the nonlinear case the breather energy isabruptly localized on the CP and its nearest and next-nearestneighbors. We present below only this latter case.

Figures 7, which show the numerical solution of Eqs. 2for x01.410, represent a typical breather motion. Figure7a1 shows that the CP moves off the initial position x 0 inthe shallower well and attempts to proceed to the deepestwell of the effective potential, but cannot do so because itskinetic energy is not sufficient to overcome the potential bar-rier. As a result, the CP remains trapped in this shallow wellduring the ensuing motion, with an amplitude of oscillations

FIG. 6. Plots showing the evolution of the positions in arb.

units of particles that make up the breather motion, for x01.414, Cl0, and Cnl1.90910

4. a1: xs(t). b1: x s1( t)

c1: xs2(t). t designates the time in arb. units. a2, b2, and

c2 show the Fourier transforms of the motion represented in a1,

b1, and c1, respectively.

TABLE IV. Characteristic parameters ke2 for the CP motion in

the trapped regime, for zero 0,0, linear (Cl,0) and nonlinear

(0,Cnl ) coupling.

(Cl ,Cnl ) ke(Cl ,Cnl )2

0,02x0

21

x02

(Cl,0) ke0,02 1 2Clx0

21x02 2x0

21/2

x021 2

(0,Cnl )ke0,0

2 1 2Cnl x03x04x0

21 x0

23 2x0

21/2

x021 2


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that is nearly half of the amplitude for a typical untrapped

regime. Moreover, the mean positions of the particles are

shifted in the direction of the shallow well, thus leading to

the presence of a dc component in the Fourier transform of

the motion of those particles see Figs. 7a2 and 7b2.

Even and odd harmonics are present in the motion contrary

to the untrapped regime with linear coupling forces, where

only odd harmonics are present in the breather motion Figs.

4a2 and 4b2, or Figs. 5a2 and 5b2. Furthermore, Figs.

7a2 and 7b2 demonstrate that the chosen initial condition

for the CP corresponds to a situation where a resonance oc-

curs, owing to the presence of the second harmonic in the

phonon band. The amplitude of oscillation for particles s

2 remains within 109, and no appreciable motion was

detected for the other particles. This behavior therefore re-

veals the ability of a -four breather to compactify in the

trapped regime.

As a general remark, for all the trapped regimes, we note

that the deviation between the CP motion in Eq. 18, and the

numerical solutions, measured by defined above, does

not exceed 5%; which corresponds to a fairly good agree-ment.

V. CRITICAL REGIME

When the CP is released from position x 0x0ec , its en-

ergy is just sufficient to overcome the potential barrier. Inthis case, the breather motion depends critically on the en-ergy transfer process from the CP to the neighboring latticesites. The dynamics can be arbitrarily divided into two maincategories of breather oscillations, which are discussed be-low for the linear coupling case only, because the dynamicalbehavior in the nonlinear coupling case is similar.

A. Trapped regime in the deepest well of the effective potential

In the presence of linear coupling forces, for x 01.411 992, Fig. 8a1 shows a particular situation in whichthe CP is initially untrapped and then becomes trapped in thedeepest well. The CP potential energy is initially sufficient topermit it to overcome the potential barrier. The CP moves up

to the deepest well of the effective potential, reaches theturning point, then begins to travel back to the shallow well,and finally has an energy that is no longer sufficient forcrossing the potential barrier. It is therefore trapped in thedeepest well of the potential during the ensuing oscillatorymotion. We attribute this trapping to the fact that, a smallfraction of CP energy is converted into energy for its neigh-bors, through a phonon emission process, as we already men-tioned above. Although this energy conversion process isvery weak in case of small coupling forces, it is sufficient toplay a crucial role in the critical regime.

B. Temporary trapping

When the initial position is x01.411 996 slightly largerthan x 0 in Fig. 8a1 the energy is just sufficient for the CPto execute several untrapped oscillations in the beginning ofits motion. In this case, Fig. 8a2 shows that the CP ex-ecutes several untrapped oscillations at one go, before under-going a transition to the trapped regime. A careful examina-tion of Fig. 8 b2 reveals that the trapping of the CP occursprecisely when the amplitude of oscillations of the nearestneighbors, s1, attains its maximum at t100. In otherwords, this trapping occurs exactly when a part of energythat was initially stored in the CP is transferred to the nearestneighbors s1 amplitude 2103. Then, after a while,the energy lost by the CP is at least in part transferred back

FIG. 7. Plots showing the particle dynamics in the trapped re-

gime of the CP, for x01.410, Cl0 and Cnl1.909104. FIG. 8. Plots showing the particle dynamics in the critical re-

gime for Cl6.5104 and Cnl0. x01.411 992: a1, b1, and

c1. x01.411 996: a2, b2, and c2.


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to the CP while this particle is moving in the direction of thepotential barrier; which permits the CP to cross again thepotential barrier. So, the CP executes several untrapped os-cillations, until a second temporary trapping occurs. We ob-serve in Fig. 8a2 that the CP undergoes several temporarytrappings in this untrapped regime. Thus, the existence of acritical regime for a Klein-Gordon breather makes the great-est qualitative difference with previous studies on the

breather dynamics in nonlinear Klein-Gordon systems. Ourresults in Figs. 8 therefore demonstrate that Klein-Gordonbreathers are collective entities that possess a much richertemporal structure of modes than that of a simple particle.

VI. CONCLUSION

We have investigated theoretically the dynamics of ahighly discrete breather in the ultimate degree of energy lo-calization in a nonlinear Klein-Gordon lattice, with weak in-terparticle coupling, submitted to a -four substrate poten-tial. In this system, the central particle CP executesrelatively large-amplitude nonlinear oscillations, while all

the other particles of the lattice execute small-amplitude os-cillations around their equilibrium positions. For this -fourbreather, we have demonstrated the existence of several re-gimes of oscillatory motion.

i We have observed an untrapped regime in which theCP executes large-amplitude oscillations from one to theother side of the potential barrier.

ii In other parameter regions, we find the trapped re-gime, in which the CP is trapped in one of the two wells ofits effective potential throughout its motion. Since the cou-pling forces breaks the symmetry of the two wells of theeffective potential of the CP, two types of trapped regimescan occur depending on whether this CP is trapped in thedeepest or shallower well of the potential. The trapped re-

gime in the deepest well corresponds to the case where allthe particles that make up the breather oscillate on the sameside of the potential barrier, whereas the trapped regime inthe well with the smallest depth corresponds to the casewhere the CP oscillates in the shallow well while all theother particles oscillate in the deepest well.

iii Between the untrappedand trappedregimes, we haveidentified a critical regime, which occurs when the CP startsout its motion in the shallower well with an energy that is

just sufficient to cross the potential barrier. Then the CPundergoes several temporary trappings within an untrappedregime. This temporary trapping occurs because energy istransferred from the CP to the nearest neighbors. However,

the nearest neighbors can synchronously transfer energyback to the CP precisely when this particle is attempting tocross the potential barrier, which then causes the CP to be-

come untrapped. This behavior result makes the greatestqualitative difference from previous work 10 and relatedstudies on breather dynamics 14,22.

On the other hand, we have shown that although the cou-pling forces between adjacent particles are very weak, in allthe regimes mentioned above, a very small fraction of thebreather energy is lost through a phonon-radiation process.In general, this energy-radiation process occurs in a continu-

ous and smooth manner, in the form of phonons that areemitted around the fundamental frequency 0 with a nearlyzero group velocity. Consequently, those phonons do notpropagate far away from the breather, thus preserving a highdegree of energy localization within the breather. However,we have found some particular conditions for which theenergy-transfer process is much stronger than in the casementioned above. The mechanism of this strong emission ofphonons involves a parametric coupling between phononmodes and the harmonics of a fundamental frequency as-sociated with the breather motion. A strong phonon emissionoccurs whenever the initial position of the CP is such that ahigher-order harmonic of the breather motion, n, falls in

the phonon band. In the situation of high degree of energylocalization, almost all the energy of the breather is locatedon the CP at least in the beginning of the motion. Then, thephonon emission process induces a small transfer of energyfrom the CP to particles lying away from the breather. Amajor result of the present paper is the demonstration that in

case of purely anharmonic coupling forces between adjacent

particles, the amplitude of oscillations of the particles neigh-

boring the CP decreases to zero beyond the next-nearest

neighbors. This behavior in a nonlinear Klein-Gordon lattice

with an on-site double-well potential reveals the ability of a

breather to compactify.In this context, the concept of compactification of solitary

waves 10,12,14,22,23 suggests the possibility of existence

of collective entities with a strict localization of energy inphysical nonlinear systems such as hydrogen-bonded chains,DNA macromolecules, or polymer chains, as mentioned inthe Introduction. This concept should be especially interest-ing for systems such as fiber transmission links in commu-nications systems, in which solitary waves with compactsupport would have as a main advantage compared to clas-sical solitons, the absence of long-range interaction betweenadjacent waves. From a fundamental point of view, an un-derstanding of the behavior of structures with a high degreeof energy localization in simple nonlinear systems consti-tutes a useful step toward the generic physical mechanismsthat govern the generation and stability of localized nonlin-

ear waves. In more complicated systems, these processesmay involve strongly anharmonic local dynamics of oneatom or ion.

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