Breathers and multibreathers

7/27/2019 Breathers and multibreathers

1/17

International Journal of Bifurcation and Chaos, Vol. 21, No. 8 (2011) 2161 2177c World Scientic Publishing Company

DOI: 10.1142/S0218127411029690

MULTIBREATHER AND VORTEX BREATHERSTABILITY IN KLEINGORDON LATTICES:

EQUIVALENCE BETWEEN TWODIFFERENT APPROACHES

J. CUEVASNonlinear Physics Group of the University of Sevilla,

Departamento de Fsica Aplicada I, Escuela Politecnica Superior,C/Virgen de Africa 7, 41011 Sevilla, Spain

[email protected]

V. KOUKOULOYANNISDepartment of Civil Engineering and Department of Mechanical Engineering,

Technological Educational Institute of Serres,62124 Serres, Greece

[email protected]

P. G. KEVREKIDISDepartment of Mathematics and Statistics,

University of Massachusetts,Amherst, MA 01003-4515, USA

[email protected]

J. F. R. ARCHILLANonlinear Physics Group of the University of Sevilla,

Departamento de Fsica Aplicada I, ETSI Inform atica,Avda. Reina Mercedes s/n, 41012 Sevilla, Spain

[email protected]

Received December 19, 2010; Revised March 28, 2011

In this work, we revisit the question of stability of multibreather congurations, i.e. discretebreathers with multiple excited sites at the anti-continuum limit of uncoupled oscillators. Wepresent two methods that yield quantitative predictions about the Floquet multipliers of thelinear stability analysis around such exponentially localized in space, time-periodic orbits, basedon the Aubry band method and the MacKay effective Hamiltonian method, and prove that bymaking the suitable assumptions about the form of the bands in the Aubry band theory, theirconclusions are equivalent. Subsequently, we showcase the usefulness of the methods through aseries of case examples including one-dimensional multi-breathers, and two-dimensional vortexbreathers in the case of a lattice of linearly coupled oscillators with the Morse potential and inthat of the discrete 4 model.

Keywords : Discrete breathers; multibreathers; stability of multibreathers.

2161
http://dx.doi.org/10.1142/S0218127411029690http://dx.doi.org/10.1142/S0218127411029690


2/17

2162 J. Cuevas et al.

1. Introduction

Over the past two decades, there has been anexplosion of interest towards the study of IntrinsicLocalized Modes (ILMs), otherwise termed discretebreathers [ Flach & Willis , 1998; Flach & Gorbach ,2008]. This activity has been, to a considerableextent, fueled by the ever-expanding applicabilityof those, exponentially localized in space and peri-odic in time, modes. A partial list of the relevantapplications includes their emergence in halide-bridged t ransition metal complexes as e.g. in [ Swan-son et al. , 1999], their potential role in the formationof denaturation bubbles in the DNA double stranddynamics summarized e.g. in [ Peyrard , 2004], theirobservation in driven micromechanical cantileverarrays as shown in [Sato et al. , 2006], their investiga-

tion in coupled torsion pendula [ Cuevas et al. , 2009],electrical transmission lines [English et al. 2008;English et al. , 2010], layered antiferromagnetic sam-ples such as those of a (C 2 H5 NH3 )2 CuCl 4 [Englishet al. , 2001b, 2001a], as well in nonlinear optics[Lederer et al. , 2008] and possibly in atomic physicsof BoseEinstein condensates [ Brazhnyi & Konotop ,2004; Morsch & Oberthaler , 2006] and most recentlyeven in granular crystals [ Boechler et al. , 2010].

In parallel to the above experimental develop-ments in this diverse set of areas, there has beenconsiderable progress towards the theoretical under-standing of the existence and stability properties of such localized modes summarized in a number of reviews and books; see e.g. [Flach & Willis , 1998;Flach & Gorbach , 2008; Lederer et al. , 2008; Aubry ,1997; Kevrekidis , 2009]. Arguably, one of the mostimportant developments in establishing the funda-mental relevance of this area in coupled nonlinearoscillator chains has been the work of MacKay andAubry [1994], which established the fact that if asingle oscillator has a periodic orbit (and relevantnonresonance conditions are satised), then upon

inclusion of a nonvanishing coupling between adja-cent such oscillators, an ILM type waveform willgenerically persist.

Given the conrmation of persistence of suchmodes, naturally, the next question concerns theirrobustness under the dynamical evolution of therelevant systems, which is critical towards theirexperimental observability. This proved to be a sub-stantially more difficult question to answer in aquantitative fashion, especially so for ILMs featur-ing multiple localized peaks, i.e. multisite breathers(since single-site breathers are typically stable in

chains of linearly coupled anharmonic oscillators).Two principal theories were proposed for address-ing the stability of such periodic orbits, whichcorrespond to discrete breather states (and foridentifying their corresponding Floquet multipli-

ers). Interestingly, these originated independentlyfrom the same pioneers that established (jointly)the existence of such modes in [ MacKay & Aubry ,1994]. In particular, the rst theory was pioneeredby Aubry in his seminal work [ Aubry , 1997] and willgo under the name Aubry Band (AB) theory, here-after. The second one is an effective Hamiltonianmethod which was introduced in a series of papersby MacKay and collaborators [ Ahn et al. , 2001;MacKay & Sepulchre , 2002; MacKay , 2004] (andwill be termed accordingly as MacKay EffectiveHamiltonian method (MEH)). The AB approachwas adapted to the stability of discrete breathersand multibreathers in the setting of KleinGordon(KG) lattices in the work of [ Archilla et al. , 2003];see also [Cuevas et al. , 2005, 2011]. The MEHapproach was applied to the same setting in a recentwork [Koukouloyannis & Kevrekidis , 2009]; see also[Koukouloyannis et al. , 2010].

Our aim in the present work is to unify thetwo methods by rmly establishing the equiva-lence of the stability conclusions of the Aubryband and MacKay effective Hamiltonian methods.Subsequently, to illustrate the usefulness and ver-satility of the methods, we apply them to arange of physically interesting chains of oscillatormodel examples, such as the Morse potential whicharises in the study of DNA bubbles [Peyrard ,2004], as well as the 4 potential which arisesin applications in dusty plasmas [ Koukouloyan-nis & Kourakis , 2007, 2009], as well as in eldtheory, particle physics and elsewhere; see e.g.the recent discussion of [ Cubero et al. , 2009] andthe earlier review [ Belova & Kudryavtsev , 1997]

and references therein. Our presentation is struc-tured as follows. In Sec. 2, we compare the twoapproaches and showcase the equivalence of theirconclusions. In particular, in Sec. 2.2 we showthat, although the original AB method pro-vides only qualitative results about the stabilityof multibreathers, if we make suitable assump-tions about the form of the bands we can also getquantitative results; i.e. we can calculate the cor-responding characteristic exponents. In Sec. 3, weillustrate the use of the method by calculating thecharacteristic exponents of multibreathers in some


3/17

Multibreather and Vortex Breather Stability in KleinGordon Lattices 2163

classic lattice congurations. In particular, we studya 1D and 2D KleinGordon lattice with Morseon-site potential (Secs. 3.1 and 3.2, respectively)and a 4 1D chain (Sec. 3.3). Finally, in Sec. 4, wesummarize our ndings and present our conclusions.

2. Comparison Between the TwoApproaches

2.1. Preliminaries Terminology

The relevant system under consideration will bea KleinGordon chain of oscillators with nearest-neighbor interaction and Hamiltonian

H = H 0 + H 1

=

i=

12 p

2

i + V (x i ) +2

i= (xi x i 1 )

2

.

(1)

As indicated previously, we will examine thetwo approaches (AB and MEH) for the linear sta-bility of multisite breathers of this general class of systems. Both approaches are based on the notionof the anti-continuum limit. In this limit ( = 0) weconsider n central oscillators moving in periodicorbits with the same frequency (this will be ourmultibreather for = 0 ), while the rest lie at

the equilibrium ( x, x) = (0 , 0). For = 0, some of these congurations, depending on the phase differ-ences between the oscillators, can be continued inorder to provide multibreather solutions. It is inter-esting/relevant to note here that while the MEHapproach provides explicit conditions about whichcongurations can be continued to nite (the criti-cal points of the relevant effective Hamiltonian), theAB theory provides only stability information for agiven conguration (for which we already know oth-erwise that it should exist at nite ).

The linear stability of these solutions is deter-mined by the corresponding Floquet multipliers.For a stable multibreather, we require that allthe multipliers lie on the unit circle. In the anti-continuum limit these multipliers lie in three bun-dles. The two conjugate ones, that correspond to thenoncentral oscillators, lie at e iT , while the thirdone lies at +1 and consists of n multiplier pairs,corresponding to the central oscillators. Each pairof +1s corresponds to the phase mode and growth mode of each isolated excited oscillator, meaningthat a small change in the initial phase or a small

change in frequency leads to an extremely close

periodic solution, for the growth mode with slightlylarger or smaller amplitude.

For = 0, the noncentral corresponding bun-dles split and their multipliers move along the unitcircle to form the phonon band, while the multipli-

ers at unity can move along, either in the unit circle(stability), or along the real axis (instability). How-ever, a pair of multipliers always remains at +1 cor-responding to the phase mode and growth mode of the whole system. Hence, the stability of the multi-breather, at least for small values of the coupling,is determined by the multipliers of the central oscil-lators. For larger values of , a HamiltonianHopf bifurcation can occur and destabilize an initiallystable multibreather.

At this point, it is relevant to make a note inpassing about the striking similarities between thediscussion above (at and near the anti-continuumlimit) and that of the linear stability of standingwaves in the discrete nonlinear Schr odinger (DNLS)equation. In that case, due to the monochromaticnature of the solutions and the U(1) invarianceof the latter model, it is possible to directly con-sider the eigenvalues associated with the standingwave solutions. However, there is a direct analogywith the spectrum of the excited sites being associ-ated with the eigenvalues at the origin at the anti-continuum limit and the continuous spectrum lying

at a nite distance from the spectral plane origin,and how at nite coupling these zero eigenvaluepairs of the excited oscillators are the ones that maygive rise to instability. In fact, it turns out that eventhe conditions under which instability will ensue formultibreathers of the KG models directly parallelthe ones for multibreathers (or multisite standingwaves) of the DNLS. The latter are analyzed in con-siderable detail for one-, two- and three-dimensionalsettings in [ Pelinovsky et al. , 2005b, 2005a; Lukaset al. , 2008]; see also [Kevrekidis , 2009].

Returning to our KG setting, the MEHapproach considers the Floquet multipliers given as = exp( T ), where T = 2 / and are the char-acteristic exponents, whereas in the AB approach, = exp(i ). Then,

=iT

=i2

. (2)

Due to the symplectic character of the Floquetmatrix if is a multiplier, so is 1 , and due toits real character if a is nonreal multiplier, so is , where the asterisk denotes the complex conju-

gate. Therefore, the corresponding multipliers come


4/17


in complex quadruplets ( , 1 , , 1 ) if || = 1and is not real, or in pairs , 1 if is real, or, if || = 1 and not real. In addition, due to thetime translation invariance of the system there isalways a pair of eigenvalues at +1. This as a result

has both i s and is, come also in quadruplets orpairs ( , ) if is real, (( , ) if is imaginary)or ( , ) if is imaginary (( , ) if is real).In principle, the pairs could collide at a single value = 1 or = 1 if the system undergoes an expo-nential instability or a period-doubling bifurcation,respectively, although there is always a pair of +1sfor the systems under study, as explained above.

2.2. The MEH approach

The MEH approach consists of constructing aneffective Hamiltonian, whose critical points are incorrespondence with the periodic orbits (in ourcase multibreathers) of the original system. Thismethod has been originally proposed in [ Ahn et al. ,2001; MacKay & Sepulchre , 2002; MacKay , 2004]and used in the present form in [ Koukouloyannis &MacKay , 2005]. The effective Hamiltonian can beconstructed as follows.

After considering the central oscillators weapply the action-angle canonical transformation tothem. Note that, in the anti-continuous limit, themotion of the central oscillators, in the action-anglevariables, is described by wi = i t + wi0 , J i =const ., for i = 0 , . . . , n 1, where wi is the angle,wi0 is the initial phase and J i the action of theith central oscillator. For this kind of systems, theaction of an oscillator can be calculated as

J i =1

2 T

0 pi dx i =

12

T

0[x i (t)]

2 dt. (3)

Since we are interested in a rst order approach,

the effective Hamiltonian can be written as H eff

=H 0 + H 1 , by neglecting terms which do not con-tribute to the results in this order of approxima-tion. In this formula, H 1 is the average value of the coupling term of the Hamiltonian, over an anglein the anti-continuous limit, which is equivalent tothe average value of H 1 over a period

H 1 =1T H 1 dt.

This averaging procedure is performed in order to

lift the phase degeneracy of the system. For the

same reason we introduce a second canonical trans-formation

= w0 , A= J 0 + + J n 1

i= w

i w

i1 , I

i=

n 1

j = iJ

j, i = 1 , . . . , n

1.

(4)

In these variables, the effective Hamiltonian reads

H eff = H 0 (I i ) + H 1 (i , I i ). (5)

Note that, since the calculations are performed inthe anti-continuous limit, the contribution of thenoncentral oscillators has disappeared.

As we seldom know the explicit form of thetransformation ( x, p) (w, J ), we use the fact thatsince the motion of the central oscillators for = 0 isperiodic, and possesses the t t, x x, p psymmetry, it can be described by a cosine Fourierseries x i(t) = k=0 Ak (J i ) cos(kwi ).

Note that at the anti-continuous limit, theorbits differ only in phase (i.e. i = i), thereforeJ i = J and the coefficients Ak s do not depend onthe index i.

So, excluding the constant terms, H 1 becomesfor the KG problem [ Koukouloyannis & Kevrekidis ,2009]

H 1 = 12

k=1

n 1

i=1A2k cos(ki ). (6)

One of the main features of the MEHapproach is that the critical points of this effec-tive Hamiltonian correspond to the multibreathersolutions of the system. This fact providesthe corresponding persistence conditions, as thesimple roots of ( H 1 / i ) = 0. Remarkably, inthis setting, similarly to what is known alsofor the DNLS [Kevrekidis , 2009], it can be provedthat the only available multibreather solutions inthe one-dimensional case are the ones with relativephase among the excited sites of 0 or .

The second important fact the MEH approachyields is that the linear stability of these criticalpoints (i.e. the Hessian of the effective Hamilto-nian) determines the stability of the correspondingmultibreather. In particular, the nonzero character-istic exponents of the central oscillators i (see thediscussion in the previous subsection) are given aseigenvalues of the stability matrix E = J D 2H eff

where J = O I

I O is the matrix of the symplectic

structure. By using the form in ( 5) for H eff

we get:


5/17


E =A B

C D=

A 1 B 1

C 0 + C 1 D 1

=

2 H 1

i I j

2 H 1

i j 2 H 0I i I j

+ 2 H 1I i I j

2 H 1 j I i

. (7)

Since the only permitted values of the relativephases are i = 0, or i = , the matrix simpli-es considerably, acquiring the form:

E =O BC O

=O B 1

C 0 + C 1 O. (8)

Then, if we consider only the dominant eigenvaluecontributions, we get that 2i = BC , where BCare the eigenvalues of the ( n 1 n 1) matrixB 1 C 0 which reads

B 1 C 0 = J

Z = J

2f 1 f 1 0f 2 2f 2 f 2 0

. . . . . . . . .

0 f n 2 2f n 2 f n 20 f n 1 2f n 1

. (9)

In this expression = H 0 /J denotes the fre-quency, while

f i f (i) =12

k=1k2 A2k cos(ki ). (10)

This leads to the characteristic exponents (i.e. effec-tive eigenvalues) of the DB in the form:

=

J

z, (11)

with z being the eigenvalues of the ( n 1n 1)matrix

Z i,j =

Z i,i 1 = f iZ i,i = 2 f i0 otherwise .

(12)

2.3. The AB approach

We demonstrate hereby that ( 11) can be reobtainedbased on the AB approach, by using the expositionof [Archilla et al. , 2003]. To this end, we recall thatthe aim of the AB approach is to look for the dis-placement that Aubrys bands [ Aubry , 1997] expe-rience when the coupling is switched on. What weplan to do below is to calculate the Floquet eigen-values assuming that the bands are parabolic andtheir shape does not change when the coupling isintroduced.

First, we recall the basics of Aubrys bandtheory with the notation used in [ Archilla et al. ,

2003] adapted to the notation in the present paper,

where it is convenient for ease of comparison.The Hamilton equations applied to the Hamiltonianof Eq. (1) can be written as:

x i + V (x i ) + H 1x i

= 0 , i = 1 , . . . , N , (13)

for a generic coupling potential H 1 , or, if i t isharmonic:

x i + V (x i ) + N

i=1C i,j x j = 0 , i = 1 , . . . , N (14)

where C is a coupling constant matrix. Let usdene x [x1 (t), . . . , x N (t)] (meaning the trans-pose matrix). Dening V (x) = [V (x1 ), . . . , V (xN )],H 1 /x = [H 1 /x 1 , . . . , H 1 /x N ] and so on,Eq. (13) can be written as:

x + V (x) + H 1x

= 0 . (15)

Suppose that x(t) is a time-periodic solution, withperiod T and frequency , its (linear) stabilitydepends on the characteristic equation for theNewton operator N given by

N (u) + V (x) + 2H 1x 2

= E, (16)

where product is the list product, i.e. f (x) isthe column matrix with elements f (x i (t)) i (t), and 2 H 1 /x 2 is the matrix of functions 2H 1 /x i x j ,

which depends on t through x = x(t).


6/17


If E = 0, this equation describes the evolutionof small perturbations = (t) of x = x(t), whichdetermines the stability or instability of x. It is how-ever, extremely useful to consider the characteristicequation for any eigenvalue E as it is the corner-

stone for Aubrys band theory.Any solution of Eq. (16) is determined bythe column matrix of the initial conditions forpositions and momenta (0) = [ 1 (0), . . . , N (0),1 (0), . . . , N (0)], with i (t) = i (t). A basisof solutions is given by the 2 N functions withinitial conditions (0), = 1 , . . . , 2N , with l (0) = l .

The Newton operator depends on theT -periodic solution x(t), and therefore, it is alsoT -periodic and its eigenfunctions can be chosenalso as eigenfunctions of the operator of translationin time (a period T ). They are the Bloch functions (i , t ) = (i , t ) exp(ii t/T ), with (, t) being acolumn matrix of T -periodic functions. The sets

{ (i , 0), (i , 0)} are also the eigenvectors of theFloquet operator F E or monodromy, that maps(0) into (T ), that is, ( T ) = F E (0). Theircorresponding eigenvalues are the 2 N multipliers{ i} = exp( i ), with {i}being the 2 N Floquetarguments.

The set of points ( , E ), with being a realFloquet argument of F E , has a band structure.As the Newton and Floquet operators are real,the Floquet multipliers come in complex conjugatepairs. Therefore, if ( , E ) belongs to a band (i.e. is real), (, E ) does as well, i.e. the bands aresymmetric with respect to , which implies thatdE/ d(0) = 0. There are always two T -periodicsolutions, with Floquet multiplier = 1 ( = 0)for E = 0. One is x(t), which represents a changein phase of the solution x(t) and corresponds to thephase mode ; the other is the growth mode , givenby x (t)/ , and represents a change in frequencyand consequently in amplitude. The consequence isthat there is always a symmetric band tangent tothe axis E = 0 at = 0.

There are at most 2 N points for a given valueof E and, therefore, there are at most 2 N bandscrossing the horizontal axes E = 0 in the space of coordinates ( , E ). The condition for linear stabilityof x(t) is equivalent to the existence of 2 N bandscrossing the axis E = 0 (including tangent pointswith their multiplicity). If a parameter like the cou-pling changes, the bands evolve continuously, andthey can lose crossing points with E = 0, leading to

an instability of the system.

The rst item to nd out are the bands at theanti-continuous limit, where Eq. ( 14) reduces to N identical equations:

x i + V (x i ) = 0 . (17)

If we consider solutions around a minimum of V ,the oscillators can be at rest x i = 0, or oscillatingwith period T ; the latter are identical except for achange in the initial phase, so they can be writtenas x i (t) = g(t + wi0 ) with g(t) being the onlyT -periodic, time-symmetric solution of Eq. ( 17)with g(0) > g (). Therefore, the excited oscillatorscan be written as:

x i(t) = z0 + 2

k=1zk cos[k(t + wi0 )]

=

k=0Ak cos[k(t + wi0 )]

=

k=0Ak cos(kwi ), (18)

with Ak = 2 zk if k > 0, A0 = z0 and wi = t + wi0 .Let n be the number of excited oscillators at

the anti-continuous limit, labeled i = 0 , . . . , n 1.Then, there are n identical bands tangent to theaxis E = 0 at = 0 for each excited oscillator,and N n bands, corresponding to the oscilla-tors at rest, with 2( N n) points intersecting theE = 0 axis.

In what follows we will show that, by mak-ing the appropriate assumptions for the excitedbands, we can use the AB method to calculatethe characteristic exponents of the multibreatherand the results are equivalent to the MEH method.These assumptions are that the form of the bandsis parabolic and remains that way in the rst orderof approximation. Thus, the excited bands can beapproximated around ( , E ) = (0 , 0) by

E () E 0 + 2 , (19)with E 0 = q and q being the eigenvalues of the(n n) Q-matrix dened below. Additionally,

=12

2E 2

= 2

42 J H

= 1

T 2J H

(20)

where we have made use of [Archilla et al. , 2003,Equation (B14)]. The factor is positive if the

on-site potential V is hard and negative if V is


7/17


soft (a potential is hard if the oscillation ampli-tude increases with the frequency and soft other-wise). When the coupling is switched on, the bandswill move and change shape; the E = 0 eigenvalueis degenerate with multiplicity N n at = 0,but this degeneracy is generically lifted for = 0and only one band will continue being tangent at(, E ) = (0 , 0) due to the phase mode. Applyingdegenerate perturbation theory to Eq. ( 16), withH 1 being the perturbation, a perturbation matrixQ can be constructed [ Archilla et al. , 2003], whoseeigenvalues q are those of the perturbed Newtonoperator. The nondiagonal elements of Q aregiven by

Qij =1

i j T

0

x i 2H 1

x i x jx j dt, i = j,

i = 0 , . . . , n 1, j = 0 , . . . , n 1, (21)

with i = T

0 (x i)2 dt. The diagonal elementsare

Qi,i = j = i

ji

Qi,j . (22)

If the on-site potential V (x i) is homogeneous andthe coupling is given as in Eq. ( 1), as is the casein the present paper, i = (2 J )1 / 2 i. Let us cal-culate the derivatives of H 1 , h i,j = 2 H 1 /x i x j .Because of the way the diagonal elements of Qare constructed, we only need the derivatives withi = j . It is easy to see that they are zero exceptfor h i 1 ,i = h i,i 1 = q i (dened below) for i =1, . . . , n 1. The derivatives h0 ,n 1 and hn 1 ,0 arealso zero as the oscillators at the extremes of themultibreather are not coupled between them. Then,the matrix Q becomes:

Qi,j =

Qi,i 1 = Qi 1 ,i = q i , for i = 1 , . . . , n 1Qi,i = q i 1 + q i , for i = 1 , . . . , n 2Q0 ,0 = q 1Qn 1 ,n 1 = q n 10 otherwise

(23)

or, explicitly:

Q =

q 1 q 1 0q 1 q 1 + q 2 q 2 0

.. .

.. .

.. .

0 q n 2 q n 2 + q n 1 q n 10 q n 1 q n 1

,

(24)

with

q i q (i) = T

0x i(t)x i 1 (t)dt

T

0[x i (t)]2 dt

=2J

k 1k2 A2k cos(ki )

=J

f i , i = 1 , . . . , n 1, (25)Then, by using [Sandstede , 1998, Lemma 5.4]

we see that the matrices Q and ( /J )Z have thesame nonzero eigenvalues i.e.

q =J

z . (26)

In addition, Q also has a zero eigenvalue. 1

Some important values of q () are the followingones:

q (0) = 1 (27)

q () = k1

(1)kk2 z2k

k 1k2 z2k

. (28)

For a Morse potential, = ; for an even poten-tial, = 1, as shown in [Archilla et al. , 2003].

According to the AB theory [ Aubry , 1997], theFloquet multipliers are given by the cuts of thebands with the E = 0 axis; thus

= E 0 = J z (29)1 The fact that the Z matrix does not have a zero eigenvalue is a key point for the MEH method. This method is based in thecontinuation from the anti-continuous limit, which means that it makes use of the implicit function theorem. The Z matrix isdirectly related to the continuation matrix. So, if Z had a 0 eigenvalue the continuation would fail since the implicit function

theorem would not be valid. This is the reason we use H eff

instead of H .


8/17


and, by using (20) we get

= T J z , (30)where we have taken into account that 2

H

=H J

J

= J

. (31)

Finally, introducing ( 30) into (2), we get (11),which completes the proof of equivalence of therelevant Floquet multiplier predictions.

3. Some Examples

In what follows we will illustrate the use of themethod in order to acquire stability results by cal-

culating the characteristic exponents of some pro-totypical KleinGordon lattice congurations. InSec. 3.1 we calculate the exponents of a multi-breather in a 1D KleinGordon chain with aMorse on-site potential. These results were alreadyknown from earlier works (e.g. [Koukouloyannis &Ichtiaroglou , 2006]) for the 2-site and 3-site cases.In the present exposition we generalize these resultsfor the n-site breather case. In Sec. 3.2, we showthat this method can also be used for calculationsin higher dimensional lattices. For this purpose weuse a 2D square lattice with Morse on-site potentialand focus on the case of vortex-breathers. Finally,in Sec. 3.3, we show how this method can be usedin cases where the analytical calculations are toocumbersome to handle. In particular, we use a 1DKleinGordon chain with hard 4 potential and therotating wave approximation in order to avoid thecomplicated calculations.

3.1. The 1D Morse KleinGordon chain

We now consider some special case examples, start-ing with a linearly coupled lattice of oscillatorssubject to the Morse potential, V (x) = (exp( x) 1)2 / 2. As indicated previously, the only congura-tions that may exist in the one-dimensional settingare ones which involve excited oscillators either in-phase (i.e. with i = 0) or out-of-phase (i.e. with

i = ); see [Cuevas et al. , 2005; Koukouloyannis &Kevrekidis , 2009] and also [Cuevas , 2003] fo r adetailed discussion. Here, we proceed to performsome explicit calculations for the Floquet multipli-ers in the case of n-site breathers. In what follows

we consider only the positive characteristic expo-nents . To this end we express ( 11) making useof (26)3 :

= J J q() (32)where q() denotes the Q-matrix eigenvalues for agiven . It is straightforward to show that 4

q(0) = 4 sin2 m

2n, m = 1 , . . . , n 1 (33)

and that

q() = q(0). (34)For instance, in the case of a 2-site breather,

q(0) = 2 and q() = 2 .We now focus on the particular case of theMorse potential, since it is a potential for whichclosed form analytical expressions can be found.[For other types of potentials, some approximationscan be made for small and high frequencies;alternatively, the required single-oscillator param-eters, such as J and /J can be calculatednumerically.]

In the Morse case and in order to evaluate J and/J , we express J as a function of the Fouriercoefficients:

J = 2 k 1

k2 z2k . (35)

For this potential, it is

z0 = ln1 + 22 ; zk =

(

1)k

k rk/ 2

, r =1

1 + .(36)

Substituting into the action

J = 1 J

= 1. (37)2 The expression = H/J comes from the Hamiltons equations for the action-angle variables, since all the calculations areperformed in the uncoupled, and therefore integrable, limit.3 In what follows, and in order to x ideas, given the equivalence of the two methods, we will use the formulation with theQ-matrix.4

We are neglecting the 0 eigenvalue, associated with m = 0.


9/17


Thus,

() = 1 q(). (38)In the case of a general phase, we can express

q() = q () q(0) with q () given by (25). In thespecial case of the Morse potential, we have

q () =2J

k 1r k cos(k). (39)

To obtain the relevant sum, we use a simplegeometric series formula that can be found e.g.in [Gradshteyn & Ryzhik , 1965], according towhich:

q () =2

J r

cos r1 2r cos + r 2

. (40)

Consequently,

() = 2r cos r1 2r cos + r 2 q(0). (41)For the relevant values of for time-reversible

multibreathers, we get:

(0) = 1 q(0)= 2 sin

m2n 1 m = 1 , . . . , n 1

(42)

() = (1 ) q(0)= 2 sin

m2n (1 ) m = 1 , . . . , n 1.(43)

Figures 1 and 2 show, respectively, the analyt-ical eigenvalue predictions (dashed lines) for stableand unstable two-site and three-site breathers withthe Morse potential and how they favorably com-pare to the corresponding numerical results (solidlines), obtained via a fully numerical linear sta-bility analysis (and corresponding computation of the Floquet multipliers). It is clear that the predic-tions are very accurate close to the anti-continuumlimit, and their validity becomes progressively lim-ited for larger values of the coupling parameter ,yet they yield a powerful qualitative and even quan-titative (in the appropriate parametric regime) tool

for tracking the stability of these localized modes.

The gures also illustrate typical proles of the cor-responding two- and three-site ILMs.

3.2. Vortices in square Morselattices

The methodology can also be extended to latticesof higher dimensionality. We consider below somebasic properties of discrete vortex breathers of dif-ferent integer topological charges, in a square 2Dlattice. Firstly, we consider square vortices over asingle plaquette of the 2D lattice with S = 1, i.e.at the anti-continuum limit, the excited sites are(0, 0), (0, 1), (1, 1) and (1 , 0) with a phase difference = / 2 between nearest neighbors. This implies aperturbation matrix given by:

Q = q 2

Q, with Q =2 1 0 11 2 1 00 1 2 1

1 0 1 2(44)

with q (/ 2) given from (40) which is evaluated as:

q 2

= (1 )2

J (1 + 2 ). (45)

This, in turn, upon use of Eq. ( 32) implies that

= i(1 ) 1 + 2 q, (46)where q are the eigenvalues of the Q matrix. Thiscorresponds to the matrix of the normal modes of a 1D chain of four linearly coupled oscillators withperiodic boundary conditions. Let us recall that fora system of n coupled oscillators, the eigenvaluesare given by:

q = 4 sin 2mn

m = 1 , . . . , n 1, (47)in addition to the 0 eigenvalue. In the present case of four oscillators with periodic boundary conditions,the nonzero eigenvalues are given by two and four,with the former being doubly degenerate. Thus, wehave for S = 1 vortices the following spectrum:

=

i(1 ) 2 1 + 2 single eigenvalue2i(1 ) 1 + 2 double eigenvalue.

(48)

which implies stability for > 0.


10/17


10 5 0 5 100

0.2

0.4

0.6

0.8

1

1.2

1.4

n

u n

( 0 )

10 5 0 5 10 1

0.5

0

0.5

1

1.5

n

u n

( 0 )

0 0.01 0.02 0.03 0.04 0.050

0.05

0.1

R e ( )

0 0.01 0.02 0.03 0.04 0.050

0.05

0.1

0.15

0.2

I m ( )

Fig. 1. (Top panels) Proles of an in-phase (left) and an out-of-phase (right) two-site breather with the Morse potential for = 0 .8 and = 0 .05. The bottom panels show the value of the characteristic exponents of the corresponding congurations,with respect to the coupling parameter . Dashed lines correspond to the predictions of the stability theorems, while solidones to full numerical linear stability analysis results.

This type of analysis can be generalized forarbitrary values of the vorticity S , leading to theconclusion that Q is the matrix of 4 S coupled oscil-lators, which implies that vortices with any integer

topological charge will be stable for > 0 inthe case of a lattice with an on-site Morse poten-tial. For instance, in the case of the S = 2 vor-tex, we obtain the explicit expressions for theeigenvalues:

=

i(1 )2 (2 21 / 2 ) 1 + 2 single eigenvalue

i(1 )2 (2 + 2 1 / 2 ) 1 + 2 double eigenvalue

i(1 ) 2 1 + 2 double eigenvalue2i(1 )

1 + 2single eigenvalue

(49)


11/17


10 5 0 5 100

0.2

0.4

0.6

0.8

1

1.2

1.4

n

u n

( 0 )

10 5 0 5 10 1

0.5

0

0.5

1

1.5

n

u n

( 0 )

0 0.005 0.01 0.015 0.02 0.025 0.030

0.05

0.1

R e ( )

0 0.005 0.01 0.015 0.02 0.025 0.030

0.05

0.1

I m ( )

Fig. 2. Same as in Fig. 1, but now for the unstable left-panel conguration of three in-phase excited sites (with two realmultiplier pairs as shown in the bottom panel) and the case of the out-of-phase, three-site right-panel conguration, which isstable only in the proximity of the anti-continuum limit.

It is important to highlight here some interest-ing differences between the above results and thecase of the DNLS (and more generally that of evenpotentials in KG chains, including the case of thehard 4 lattice considered below). In the latter class

of problems, the vanishing of the odd coefficients inthe Fourier expansion of the periodic orbit leads tothe conclusion that q (/ 2) = 0 and hence there isno contribution of the eigenvalues to leading order.This is the situation which has been characterizedas super-symmetric in [ Pelinovsky et al. , 2005a;Kevrekidis , 2009] and one in which the higher ordercontributions are critical in determining the sta-bility. Nevertheless, in the case considered herein,the asymmetry of the Morse potential produces anonvanishing of q (/ 2) and offers a corresponding

nonzero leading order correction to the eigenvaluesat O( 1 / 2 ).

Figure 3 shows the dependence of stabilityeigenvalues for the S = 1 and S = 2 vortices andtheir comparison with the fully obtained numerical

linear stability results as a function of the coupling. As can be observed in the gures, the approx-imation is less accurate in this case, although itis qualitatively correct. The reason for the partialdisparity is that higher-order contributions to therelevant eigenvalues (whose calculation is consider-ably more technically involved) lead to the observedsplitting of all the doubly degenerate eigenvaluepairs. In the relevant cases, the analytical (dashedline) predictions can be seen to straddle the twoobserved numerical pairs.


12/17


0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.02

0.04

0.06

0.08

0.1

0.12

I m ( )

0 0.002 0.004 0.006 0.008 0.010

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

I m ( )

(a) (b)

Fig. 3. The characteristic exponents of vortex congurations with (a) S = 1 and (b) S = 2, with respect to , for the Morsepotential and = 0 .8. Dashed lines correspond to the theoretical predictions based on Eqs. ( 48) and (49), respectively; thefull numerical linear stability results are given by solid lines and indicate that all doubly degenerate eigenvalue pairs split dueto higher order contributions in the relevant expansions for the coupling constant .

3.3. The 1D hard 4 KleinGordon chain

The time evolution of a single oscillator in the hard4 potential, V (x) = x2 / 2 + x4 / 4 is given by:

x(t) = 2m1 2m cnt

1 2m, m

= 2m1 2m cn2K (m)

t,m , (50)

where cn is a Jacobi elliptic function of modu-lus m and K (m) is the complete elliptic integralof the rst kind dened as K (m) =

/ 20 [1

m sin2 x] 1 / 2 dx.The breather frequency is related to the mod-

ulus m through:

=

2 1

2mK (m). (51)

The elliptic function can be expanded into aFourier series leading to [ Abramowitz & Stegun ,1965]:

z2 +1 =

K (m) 21 2mq +1 / 2

1 + q 2 +1,

= 0 , 1, 2, . . . . (52)

where q is the elliptic Nome which is dened as

q

q (m) = exp K (1 m)

K (m). (53)

In order to get q() and /J , we cannotuse (10) and (25) because it is not possible to nd aclosed form expression. Instead, we use the integralexpression:

f (i) =1

2 T

0x i (t)x i+1 (t) d t. (54)

After some manipulations (where it is crucialto apply [Khare et al. , 2003, identity 171]), weobtain:

f () =8K (m)

3 (1 2m)[cs(a, m )ns( a, m )[2E (m)

K (m)(1 + dn 2 (a, m ))]] 8K (m)

3 (1 2m)[ds(a, m )(cs 2 (a, m ) + ns 2 (a, m ))Z (a, m )]

(55)

where E (m) is the complete elliptic integral of the second kind dened as E (m) =

/ 20 [1

m sin2 x]1 / 2 dx, Z (a, m ) is the Jacobi zeta functionand a = 2 K (m)/ .

For the action J , a similar manipulationleads to

J =16K (m)

321 m

1 2mK (m) E (m) . (56)

The derivative of this expression is cumber-some to handle. So, in what follows, we will work

instead with numerically obtained values of J and


13/17


1 1.5 2 2.5 30

2

4

6

8

10

12

14

16

J

1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

/ J

(a) (b)

Fig. 4. Dependence with respect to of the action (a) and /J (b) for the hard 4 potential. The dashed line correspondsto the prediction of the RWA [Eq. ( 59)], while the solid one represents the exact numerical result.

10 5 0 5 10 0.5

0

0.5

1

1.5

2

2.5

3

3.5

n

u n

( 0 )

10 5 0 5 10 4

3

2

1

0

1

2

3

4

n

u n

( 0 )

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

I m ( )

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

R e ( )

Fig. 5. (Top panels) Proles of an in-phase (left) and an out-of-phase (right) two-site breather with the hard 4 potential; = 3 and = 0 .05. The bottom panels show the dependence of the characteristic exponents , of the corresponding congu-rations, on the coupling parameter . The dashed lines correspond to the predictions of the stability theorems and dash-dotted

lines to the RWA predictions, while the solid ones represent the full numerical result.


14/17


/J which are relevant for time-reversible multi-breathers and vortex breathers, as for these caseswe need f (0), f () and f (/ 2). As indicated pre-viously, for every even potential, z2 +1 = 0, and,consequently, f (0) = J/ , f () = f (0) andf (/ 2) = 0. This leads to:

(0) = J J q(0), (57)() = J J q(0). (58)

Figure 4 shows the dependence of J and /J with respect to the frequency. Figures 5 and 6 illus-trate subsequently the relevant stability eigenval-ues for two-site and three-site breathers as obtained

from the expressions above and compare them tothe full numerical linear stability results. The agree-ment in this case is very good (there are no degen-eracies and associated higher-order contributionsthat may deteriorate the quality of the agreement as

in the vortex breather case above); in fact, in someof the cases, the curves are almost indistinguishablethroughout the considered parameter range.

An important observation concerns, however,the role of the hard nature of the potential. Inparticular, as illustrated in Fig. 4, the quantity/J is positive in this case, i.e. its sign is oppo-site from the soft case of the Morse potential (where/J = 1). This results in the correspondingreversal of the stability conclusions in Figs. 5 and 6,in comparison with Figs. 1 and 2 of the Morsecase. That is, in-phase modes are now stable, while

10 5 0 5 10 0.5

0

0.5

1

1.5

2

2.5

3

3.5

n

u n

( 0 )

10 5 0 5 10 4

3

2

1

0

1

2

3

4

n

u n

( 0 )

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

I m ( )

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

R e ( )

Fig. 6. Same as Fig. 5, but for the three-site in-phase (left) and out-of-phase (right) conguration. Again the dash-dotted linesin the bottom panels represent the (fully-analytical) RWA predictions, which agree well with the semi-analytical dependences

(dashed lines) and, in turn, with the full numerical results (solid lines).


15/17


out-of-phase ones are unstable (as is true for thedefocusing nonlinearity DNLS case also), while thereverse was true in the Morse potential (as well asfor the focusing DNLS case). Lastly, we recall thatsince this is an even potential and thus f (/ 2) = 0,

the leading order calculation would yield a vanish-ing contribution to the eigenvalues for the vortexcase and a higher-order calculation is necessary todetermine the stability of the latter.

As an aside towards obtaining a fully analyticalprediction for this case (as some of the quantitiesneed to be obtained numerically above), we notethe following. Although we cannot acquire an exactform for J (), as in the case of the Morse poten-tial, an approximate form for J can be found byusing the rotating wave approximation (RWA), i.e.by supposing that x(t)

2z1 cos(t). The intro-

duction in the dynamical equations for the singleoscillator leads to:

z1 = 2 13 . (59)Thus, J = 2 z21 = 2 (2 1)/ 3 and /J =1/ [2(2 1/ 3)], and the corresponding expressionsfor the eigenvalues read:

(0)

2 132

1

q(0) (60)

() 2 132 1 q(0) . (61)A comparison between the numerically

acquired values of J () and /J with the onescalculated from the RWA is shown in Fig. 4. Theagreement is remarkable and attests to the qualityof the single frequency rotating wave approxima-tion. In Figs. 5 and 6 the characteristic exponentscalculated numerically (solid lines) as well as usingEqs. ( 57) and (58) (dashed lines) and via Eqs. ( 60)and ( 61) are compared, illustrating the excellentagreement between all three.

4. Conclusions and Perspectives

The results presented in this work underscore theformulation of a toolbox that enables the sys-tematic characterization of both the qualitativeand even the quantitative aspects of stability of multibreather and vortex breather waveforms inthese large number of degrees of freedom, Hamil-

tonian lattice systems of the KleinGordon variety.

A systematic calculation of the correspondingFloquet multipliers is presented and highlights thecrucial components that imply stability, namelythe proper combination of the sign of the cou-pling constant, the nature (hard or soft) of the

potential and the relative phases between the adja-cent excited sites. For example, for positive cou-plings, and soft potentials, out-of-phase structuresmay be stable near the vanishing coupling limit,while in-phase ones are always unstable, resemblingthe corresponding DNLS predictions [ Kevrekidis ,2009]; the nature of the conclusions is reversed foreither (small) negative couplings or for hard poten-tials. The explicit analytical predictions have beentested against numerical results both for symmetric(such as the hard 4 ) and asymmetric (such as theMorse) potentials, both for hard and soft ones, andboth for simpler, nondegenerate one-dimensionalmultibreather settings and for more complex anddegenerate two-dimensional vortex breathers. In allcases, the two theories whose results were shown tobe equivalent herein, namely the Aubry band the-ory and the MacKay Effective Hamiltonian methodyield excellent qualitative and good quantitativeagreement with the full numerical linear stabilityresults. The accuracy of the predictions is lowered indegenerate cases where higher-order contributionsmay be critical in breaking the relevant degeneracy

(as we saw in the case of the discrete vortices forthe Morse model).Naturally, a number of interesting directions

for future consideration hereby arise. Perhaps thecanonical one among them would involve a system-atic derivation of higher-order corrections for pro-totypical cases where the leading order approachyields vanishing results. For instance, the charac-terization of the stability of discrete vortices in thesuper-symmetric case of phase difference = / 2for even potentials would be a natural example.Another possibility that is also emerging and wouldbe relevant to consider from a mathematical pointof view would be to examine models with inter-sitenonlinearities, such as those of the FermiPastaUlam type. In these cases, where the potential isa function V (xn xn 1 ), it is relevant to pointout that upon consideration of the so-called strainvariables r n = xn xn 1 , the problem is revertedto an on-site potential case, for which it wouldbe worthwhile to explore methods similar to theones analyzed herein. These directions are presentlyunder consideration and will be reported in future

publications.


16/17


Acknowledgments

P. G. Kevrekidis gratefully acknowledges supportfrom NSF-DMS-0349023 (CAREER), NSF-DMS-0806762, NSF-CMMI-1000337, from the Alexandervon Humboldt Foundation and from the Alexan-der S. Onassis Public Benet Foundation. J. Cuevasand J. F. R. Archilla acknowledge nancial supportfrom the MICINN project FIS2008-04848.

References

Abramowitz, M. & Stegun, I. A. [1965] Handbook of Mathematical Functions (Dover, NY).

Ahn, R., MacKay, R. S. & Sepulchre, J.-A. [2001]Dynamics of relative phases: Generalised multi-breathers, Nonlin. Dyn. 25 , 157.

Archilla, J. F. R., Cuevas, J., S anchez-Rey, B. & Alvarez,A. [2003] Demonstration of the stability or instabil-ity of multibreathers at low coupling, Physica D 180 ,235.

Aubry, S. [1997] Breathers in nonlinear lattices: Exis-tence, linear stability and quantization, Physica D 103 , 201.

Belova, T. I. & Kudryavtsev, A. E. [1997] Solitons andtheir interaction in classical eld theory, Phys. Usp.40 , 359.

Boechler, N., Theocharis, G., Job, S., Kevrekidis, P. G.,Porter, M. A. & Daraio, C. [2010] Discrete breathersin one-dimensional diatomic granular crystals, Phys.Rev. Lett. 104 , 244302.

Brazhnyi, V. A. & Konotop, V. V. [2004] Theory of non-linear matter waves in optical lattices, Mod. Phys.Lett. B 18 , 627.

Cubero, D., Cuevas, J. & Kevrekidis, P. G. [2009]Nucleation of breathers via stochastic resonance innonlinear lattices, Phys. Rev. Lett. 102 , 205505.

Cuevas, J. [2003] Localization and energy transfer innon-homogeneous anharmonic lattices, PhD thesis,University of Seville, Spain.

Cuevas, J., Archilla, J. F. R. & Romero, F. R. [2005]Effect of the introduction of impurities on the

stability properties of multibreathers, Nonlinearity 18 , 76.

Cuevas, J., English, L. Q., Kevrekidis, P. G. & Ander-son, M. [2009] Discrete breathers in a forced-dampedarray of coupled pendula: Modeling, computation andexperiment, Phys. Rev. Lett. 102 , 224101.

Cuevas, J., Archilla, J. F. R. & Romero, F. R. [2011]Stability of non-time-reversible phonobreathers, J.Phys. A.: Math. Theor. 44 , 035102.

English, L. Q., Sato, M. & Sievers, A. J. [2001a] Mod-ulational instability of the nonlinear uniform spinwave mode in easy axis antiferromagnetic chains:II. Inuence of sample shape on intrinsic localized

modes and dynamic spin defects, Phys. Rev. B 67 ,024403.

English, L. Q., Sato, M. & Sievers, A. J. [2001b]Nanoscale intrinsic localized modes in an antiferro-magnetic lattice, J. Appl. Phys. 89 , 6706.

English, L. Q., Basu Thakur, R. & Stearett, R. [2008]Patterns of travelling ilms in a driven electrical line,Phys. Rev. E 77 , 066601.

English, L. Q., Palmero, F., Sievers, A. J., Kevrekidis,P. G. & Barnak, D. H. [2010] Traveling and station-ary intrinsic localized modes and their spatial controlin electrical lattices, Phys. Rev. E 81 , 046605.

Flach, S. & Willis, C. R. [1998] Discrete breathers,Phys. Rep. 295 , 182.

Flach, S. & Gorbach, A. V. [2008] Discrete breathers Advances in theory and applications, Phys. Rep.467 , 1.

Gradshteyn, I. S. & Ryzhik, I. M. [1965] Table of Inte-

grals, Series, and Products (Academic Press, NY).Kevrekidis, P. G. [2009] The Discrete Nonlinear Schr odinger Equation (Springer-Verlag, Berlin).

Khare, A., Lakshminarayan, A. & Sukhatme, U.[2003] Local identities involving elliptic functions,ArXiV:math-ph/0306028. This paper was publishedas Pramana 62 (2004) 1201, but the referred identityonly appears in the preprint.

Koukouloyannis, V. & MacKay, R. S. [2005] Existenceand stability of 3-site breathers in a triangular lat-tice, J. Phys. A: Math. Gen. 38 , 1021.

Koukouloyannis, V. & Ichtiaroglou, S. [2006] A sta-bility criterion for multibreathers in KleinGordonchains, Int. J. Bifurcation and Chaos 16 , 18231827.

Koukouloyannis, V. & Kourakis, I. [2007] Existence of multisite intrinsic localized modes in one-dimensionaldebye crystals, Phys. Rev. E 76 , 016402.

Koukouloyannis, V. & Kevrekidis, P. G. [2009] On thestability of multibreathers in KleinGordon chains,Nonlinearity 22 , 2269.

Koukouloyannis, V. & Kourakis, I. [2009] Discretebreathers in hexagonal dusty plasma lattices, Phys.Rev. E 80 , 026402.

Koukouloyannis, V., Kevrekidis, P. G., Law, K. J. H.,

Kourakis, I. & Frantzeskakis, D. J. [2010] Existenceand stability of multisite breathers in honeycomb andhexagonal lattices, J. Phys. A: Math. Theor. 43 ,235101.

Lederer, F., Stegeman, G. I., Christodoulides, D. N.,Assanto, G., Segev, M. & Silberberg, Y. [2008] Dis-crete solitons in optics, Phys. Rep. 463 , 1.

Lukas, M., Pelinovsky, D. & Kevrekidis, P. [2008]LyapunovSchmidt reduction algorithm for three-dimensional discrete vortices, Physica D 237 ,339.

MacKay, R. S. & Aubry, S. [1994] Proof of the exis-tence of breathers for time-reversible or Hamiltonian


17/17


networks of weakly coupled oscillators, Nonlinearity 7 , 1623.

MacKay, R. S. & Sepulchre, J.-A. [2002] EffectiveHamiltonian for traveling discrete breathers, J.Phys. A: Math. Gen. 35 , 3985.

MacKay, R. S. [2004] Slow manifolds, Energy Local-isation and Transfer (World Scientic, Singapore),pp. 149192.

Morsch, O. & Oberthaler, M. O. [2006] Dynamics of BoseEinstein condensates in optical lattices, Rev.Mod. Phys. 78 , 179.

Pelinovsky, D. E., Kevrekidis, P. G. & Frantzeskakis, D.J. [2005a] Persistence and stability of discrete vor-tices in nonlinear Schrodinger lattices, Physica D 212 , 20.

Pelinovsky, D. E., Kevrekidis, P. G. & Frantzeskakis, D.J. [2005b] Stability of discrete solitons in nonlinearSchr odinger lattices, Physica D 212 , 1.

Peyrard, M. [2004] Nonlinear dynamics and statisticalmechanics of DNA, Nonlinearity 17 , R1.

Sandstede, B. [1998] Stability of multiple-pulse solu-tions, Trans. Am. Math. Soc. 350 , 429.

Sato, M., Hubbard, B. & Sievers, A. J. [2006] Nonlinearenergy localization and its manipulation in microme-chanical oscillator arrays, Rev. Mod. Phys. 78 , 137.

Swanson, B. I., Brozik, J. A., Love, S. P., Strouse, G. F.,Shreve, A. P., Bishop, A. R., Wang, W.-Z. & Salkola,M. I. [1999] Observation of intrinsically localizedmodes in a discrete low-dimensional material, Phys.Rev. Lett. 22 , 3288.

Documents

Breathers and multibreathers