Physics Letters A 377 (2013) 1543–1553

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Physics Letters A

www.elsevier.com/locate/pla

Multibreather stability in discrete Klein–Gordon equations:Beyond nearest neighbor interactions

Zoi Rapti

Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 December 2012Received in revised form 5 April 2013Accepted 18 April 2013Available online 23 April 2013Communicated by A.R. Bishop

Keywords:MultibreathersDiscrete Klein–Gordon equationsStability

We present results on multibreather stability in one-dimensional nonlinear Klein–Gordon chains. Ouranalysis is based on Aubry’s band theory and perturbation theory. First, we provide an alternative proofof the stability of multibreathers in a chain with nearest neighbor interactions only. Then, we extendour analysis to the case of interactions with up to three neighbors. For next-nearest neighbor and third-nearest neighbor interactions we also extend the theory to study the stability properties of recently foundmultibreathers that have nonstandard phase shifts (not equal to 0 or π ).

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Discrete multibreathers are time periodic solutions which are localized in space (decay to zero at infinite distance). They arise due tothe interaction of nonlinearity and discreteness. Since the work in [1] on intrinsic localized modes in anharmonic lattices there have beenextensive studies related to their existence and stability properties.

In [2] a mathematical proof of breather existence in Klein–Gordon chains was given for small coupling in a general class of oscillatornetworks. The persistence of solutions of fixed period after was formulated as a problem of finding zeros of an operator in a Banach spaceusing the implicit function theorem and then it was proved that they decay exponentially in space. In [3] the existence of multibreatherswas proved in the case of weak coupling using as starting point the limit where the oscillators are not coupled and the periodic solutionscan be found by phase portrait techniques. Then, these trivial solutions were continued as a function of the small coupling constant usingthe implicit function theorem. In the same work, the linear stability of multibreathers was considered. The main idea was to study theNewton operator obtained by the linearized system, and since the solutions are time periodic, the linear stability was performed usingFloquet theory. The Aubry band theory refers to the reformulation of Krein theory in terms of the properties of the band structure of theNewton operator. More recent work on multibreather stability using the band method of Aubry and can be found in [4]. The issue ofthe linear stability of multibreathers was studied by applying degenerate perturbation theory to Aubry’s band theory. The approach involvesmodifying the coupling matrix and knowledge of the hardness/softness of the onsite potential and the multibreather type.

In [5] small Hamiltonian perturbations of Hamiltonian oscillator networks were considered and an effective Hamiltonian on the sub-manifold of periodic orbits was introduced. The critical points of the effective Hamiltonian subject to the action (which is preserved)correspond to exact periodic solutions. The linearization of the effective Hamiltonian about the critical points gives the linearized dynam-ics of the full system to leading order in the perturbation. Hence, the linear stability of periodic orbits is determined by the effectiveHamiltonian. The stability of multibreathers was also studied in [6] using the effective Hamiltonian method of MacKay. A direct counting re-sult was given for the number of real (stable) and imaginary (unstable) characteristic exponents for arbitrary breather configurations withadjacent excited sites. In subsequent work [7], the equivalence between the band method of Aubry and the effective Hamiltonian methodof MacKay was proved. The results were illustrated using a Morse type onsite potential in Klein–Gordon chains and also in the case of thediscrete φ4 model. The full numerical linear analysis stability and the predictions of the stability theorems showed great agreement forsmall values of the coupling parameter.

The spectral stability of discrete breathers with holes (oscillators at rest between excited oscillators) has been recently studied in [8]using perturbation expansions from the zero coupling case and Floquet theory. This approach was used to also confirm the stability results

E-mail address: zrapti@illinois.edu.

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2013.04.035

1544 Z. Rapti / Physics Letters A 377 (2013) 1543–1553

in [4] and it has been used in [9] to study the multibreather stability in magnetic metamaterials. Most recently in [10] the stability ofmultibreathers in Klein–Gordon equations with anharmonic nearest neighbor coupling was investigated. Continuation from the case ofanti-continuous (zero coupling) was used to prove the existence of multibreathers. It was also proved that the stability depends on thephase difference and the distance between the two sites in each pair of excited sites in the anti-continuous solution. As in previous works[4,8] the hardness (softness) of the onsite potential and the sign of the coupling parameter (attractive or repulsive) was found to affectthe stability result.

Our motivation to study the stability of multibreathers in Klein–Gordon chains stems from DNA dynamics. These excitations are be-lieved to match experimentally observed bubbles propagating along the helix [11]. We adopt the formalism of the Peyrard–Bishop (PB)model [12], where the onsite potential is of Morse type and the coupling is harmonic. Recent experimental and numerical evidence sug-gests that interactions with neighbors that are several neighbors away occur [13]. The helical structure of the DNA forces nucleotides thatare farther part in the one-dimensional models to be close enough in the three-dimensional structure. Hence, to take into account theseinteractions a modified PB model has been considered in several works, see [14], and in particular [15] where breather dynamics wereconsidered in a modified model with coupling both between the nearest neighbor and the neighbor that is located 4 bases away. In [16]long-range dipole interactions with distant base pairs were taken into account and the existence and stability of breathers were studiednumerically. Longer range interactions have also been considered in other systems, for instance in the discrete nonlinear Schrödingerequation (DNLS) [17], where the modulational instability of plane wave solutions was considered. In [18] the effect of interactions withnon-nearest neighbors on the existence and stability of solutions was analyzed in the context of the DNLS again. Quite recently Klein–Gordon chains with interactions with farther neighbors were investigated in [19]. Using the effective Hamiltonian approach of MacKay,new multibreather solutions with nonstandard phase shifts (not equal to 0 or π ) were found in the case of interactions with the secondand third neighbors and their stability was investigated. In this work we will focus on analytical calculations, since good agreement be-tween numerical and analytical results has already been established in previous works, [4] and [19], for the model and methods that areconsidered here.

This rest of this Letter is organized as follows. In Section 2 we give the necessary terminology and notation for the problem. InSection 3.1 we provide an alternative proof to the stability properties of multibreathers in the case of nearest neighbor interaction only.In Section 3.2 we show that all multibreathers with oscillators in-phase are unstable for interactions with up to second-nearest neighborsand we provide an analytical proof to the stability results in the newly found multibreathers in [19]. In Section 3.3 we investigate thestability properties of multibreathers in the case with coupling between the nearest, second-nearest, and third-nearest neighbors. Finally,in Section 4 we present a summary of our results and conclusions.

2. Background

2.1. Discrete Klein–Gordon chains

In this work we focus on Klein–Gordon chains which include interactions with neighbors up to three sites away. The dynamicalequations of the model can be written as

yn + V ′(yn) + ε

N∑m=1

Cnm ym = 0, (1)

where yn are functions of time, ˙ denotes the derivative with respect to time, the onsite potential is of Morse type V (yn) = 12 (e−yn − 1)2,

ε is the small coupling parameter, and Cnm , with

Cnm = 0, if |n − m| > 3, n,m = 1, . . . , N

– where N is the number of oscillators in the chain – is the coupling constant between oscillators m and n. We use the notation

y(t) = [y1(t), y2(t), . . . , yN(t)

]T,

and define V (y) = [V (y1), V (y2), . . . , V (yN)]T, and analogously its derivatives. We will use C to denote the N × N coupling matrix withelements Cnm .

We assume that yn are periodic functions with period T and frequency ω. The linear stability properties of these solutions can beobtained by studying the Newton operator Nε defined by

Nεξ ≡ ξ + V ′′(y) · ξ + εCξ = Eξ, (2)

where ξ is a small perturbation of y and · is the list product, namely f (y) · ξ is the vector with elements f (yn)ξn . The Newton operatoris periodic, so Floquet–Bloch theory is the natural theoretical framework for the stability analysis. The Floquet matrix FE is the matrixobtained by integrating equations (2) 2N times from t = 0 to t = T with initial conditions Ξ j(0) = (ξ1(0), . . . , ξN (0), ξ1(0), . . . , ξN (0)),1 � j � N satisfying Ξ

jk = δkj . The equation with E = 0 describes the evolution of small perturbations ξ of y. The reason why we consider

the eigenvalue problem (2), rather than focus on the E = 0 case, is because following Aubry’s band theory [3,4], information about theeigenvalues of the Floquet matrix (Floquet multipliers) can be obtained from the properties of the Newton operator with E �= 0.

2.2. Aubry’s band theory

Due to the Hamiltonian structure of the problem the Floquet matrix is symplectic, so linear stability is equivalent to having all Floquetmultipliers on the unit circle. Equivalently, if we write eiθ for the Floquet multiplies, then the Floquet arguments θ must be real in order

Z. Rapti / Physics Letters A 377 (2013) 1543–1553 1545

to have linear stability. In [3] Aubry showed in Theorem 6 that a Floquet multiplier eiθ on the unit circle corresponds to a zero of E(θ) ofthe spectrum of the Newton operator Nε and that the converse is also true. The set of points (θ, E) with θ ∈R has a band structure.

Also, as argued in [4] due to the symplectic properties of FE if (θ, E) belongs to a band, so does (−θ, E), and consequently the functionE(θ) is symmetric (E(θ) = E(−θ)). This symmetry in turn implies that dE

dθ(0) = 0. Since y corresponds to the argument θ = 0, there is

always a band tangent to the θ -axis at θ = 0. There are at most 2N points for a given value of E , and therefore, there are at most 2Nbands crossing the θ -axis in the (θ, E) plane. Hence, it follows that a time-periodic solution is linearly stable if and only if the number ofintersection points of the bands E of the Newton operator (counted with multiplicity) with the θ -axis is 2N . When the coupling parameterε becomes nonzero, the bands also vary continuously, and they can lose crossing points with E = 0, resulting in instability. All the tangentbands at (θ, E) = (0,0) except one, move upwards or downwards.

It has been shown in [4] that the curvature d2 E/dθ2 of the band is negative for hard and positive for soft potentials. Thus, in the caseof the (soft) Morse potential the band is tangent to the E = 0 axis and concave upwards. So, instability ensues when the band movesupward and a double tangent point with the axis E = 0 is lost.

2.3. Archilla’s perturbation theory

In [4] degenerate perturbation theory [20] was used to determine the Newton operator eigenvalues E for nonzero coupling ε . The un-perturbed operator is N0ξ = ξ + V ′′(y) ·ξ and expanding Nε given in (2) in terms of ε yields (to first order in ε) the perturbation operatorN1ξ = Cξ + V ′′′(y) · yε · ξ , where y is evaluated at ε = 0 and yε = ∂ y

∂ε |ε=0. Then using the inner product 〈ξ1, ξ2〉 = ∑Nn=1

∫ T /2−T /2 ξ∗

1nξ2n dt ,where ∗ denotes the complex conjugate, one obtains the eigenvalues of N0 + εN1 in terms of the eigenvalues E0 of the unperturbedoperator N0. They are given by the relation E0 + ελi , where λi are the eigenvalues of the perturbation matrix Q with elementsQ nm = 〈yn,N1 ym〉. Here, yi , i = 1, . . . , M , denote the elements of the orthonormal basis of eigenfunctions of N0 with respect to 〈·,·〉.M is the number of excited oscillators and N − M is the number of oscillators at rest. If one denotes by y0 the identical excited oscillatorsat ε = 0, then the basis is given by

yn = 1

μ

[0, . . . , y0(t + tn),0, . . . ,0

]T,

where μ =√∫ T

2

− T2( y0(t + tn))2 dt =

√∫ T2

− T2( y0(t))2 dt and tn is phase difference between yn(t) and y0(t): yn(t) = y0(t + tn). Then, using

the properties of the periodic solution y0 describing the unperturbed oscillators and performing the integrations in the definitions can becalculated that the elements of the perturbation matrix Q are

Q nm = 0, if oscillator n or m is at rest (σnσm = 0);Q nm = Cnm, if σnσm = 1, n �= m;Q nm = −γ Cnm, if σnσm = −1, n �= m.

The diagonal elements are

Q nn = −∑m �=n

Q nm.

The convention for the code describing the multibreather is the following: σ = 0 represents an oscillator at rest yn(t) ≡ 0, σ = 1 representsan excited oscillator yn(t) = y0(t), and σ = −1 represents an excited oscillator with a phase difference of π , with the previous onesyn(t) = y0(t + T

2 ).The Morse potential is non-symmetric, so for excited oscillators that are out-of-phase, the perturbation matrix will depend on the

symmetry coefficient γ [4] defined as follows

γ = −∫ T

2

− T2

y0(t) y0(t + T2 )dt

∫ T2

− T2( y0(t))2 dt

. (3)

It was shown in [4] that γ = ω for the Morse potential and γ = 1 for a symmetric potential. The reason is because by phase-space analysisone can see that the potential dictates the shape of the periodic solutions with a given frequency, and for a symmetric potential it holdsy0(t + T

2 ) = −y0(t).

3. Stability analysis

3.1. Nearest neighbors

In [4] it was stated that “according to numerical calculations of the eigenvalues of the perturbation matrices corresponding to groupswith different codes, the number of negative and positive eigenvalues are equal to the numbers of −1 and +1 in {σnσn+1}N ′−1

n=1 ”, whereN ′ is the size of the multibreather. In that case the Klein–Gordon system with nearest neighbor linear coupling was considered, withHamiltonian

H =N∑ m

2y2

n + V (yn) + 1

2ε(yn − yn−1)

2. (4)

n=1

1546 Z. Rapti / Physics Letters A 377 (2013) 1543–1553

Here, we will present a proof of the above claim based on Sturm-theory [21] and in particular we will follow the ideas of Givens [22].An analogous result was proved recently [6] using the formulation of MacKay’s effective Hamiltonian [5] and is also contained in [8].

We begin by noticing that the perturbation matrix for a multibreather of size M + 1 has the form

Q =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−a1 a1 0 0 · · · 0 0 0a1 −a1 − a2 a2 0 · · · 0 0 00 a2 −a2 − a3 a3 · · · 0 0 0...

......

.... . .

......

...

0 0 0 0 · · · −aM−2 − aM−1 aM−1 00 0 0 0 · · · aM−1 −aM−1 − aM aM

0 0 0 0 · · · 0 aM −aM

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

where a j = −1 if σnσn+1 = 1 and a j = γ if σnσn+1 = −1. We will only focus on breathers without holes (namely without oscillators atrest between excited oscillators).

Next, we define the polynomial

P M+1(λ) = det(Q − λI),

which is the characteristic polynomial of Q . It holds

P M+1(λ) = −(λ + aM) f M(λ) − a2M f M−1(λ), (5)

where the sequence of polynomials f j is defined inductively through

f j(λ) = −(λ + a j + a j−1) f j−1(λ) − a2j−1 f j−2(λ), j = 2, . . . , M, (6)

with

f1(λ) = −(λ + a1), f0(λ) ≡ 1. (7)

By strict induction it can be shown that

f j(0) = (−1) ja1 · · ·a j, for j = 1, . . . , M, (8)

or equivalently

f j(0) = −a j f j−1(0), for j = 1, . . . , M, (9)

while

P M+1(0) = 0, (10)

and the zero eigenvalue is simple since (as shown in Appendix A.1)

P ′M+1(0) = (−1)M+1(M + 1)a1a2 · · ·aM �= 0.

The following results are true.

• No two successive polynomials have a common zero, because then all of them would have the same zero by (6). This leads to acontradiction since f0 ≡ 1.

• By mathematical induction one can prove the separation property of the zeros of consecutive polynomials. Clearly, the simple zeroof f1(λ) = −λ − a1 separates the two simple zeros of f2(λ) = λ2 + (2a1 + a2)λ + a1a2. Assume that the i − 2 simple zeros λi−2

k ,

1 � k � i − 2 of f i−2 separate the i − 1 simple zeros λi−1k , 1 � k � i − 1 of f i−1. Now, from (6) at each zero of f i−1, the sign of f i is

the opposite to the sign of f i−2. But, by the induction hypothesis, f i−2 changes sign between each pair of neighboring zeros of f i−1.Therefore, so does f i and has a zero between each neighboring pair of zeros of f i−1. In total there are i − 2 zeros of f i between thei − 1 zeros of f i−1. Also, f i(−∞) = +∞ and f i(λ

i−11 ) = −a2

i−1 f i−2(λi−11 ) < 0, so, f i has a zero to the left of the smallest zero λi−1

1

of f i−1. This is true because the sign of f i−2(λi−11 ) has to be positive, otherwise f i−2 would have another zero to the left of λi−1

1 ,

which is a contradiction, since it can only have i − 2 zeros. Similarly, f i(+∞) = f i−2(+∞) = (−1)i∞ and f i(λi−1i−1) = −a2

i−1 f i−2(λi−1i−1),

so for even i we have f i−2(λi−1i−1) > 0 and f i(λ

i−1i−1) < 0, while the opposite is true for odd i. In either case, there is a zero of f i to the

right of the largest zero λi−1i−1 of f i−1. Since f i has i zeros, the claim is proved.

Theorem 1. Multibreathers with all oscillators in-phase (out-of-phase) are unstable (stable).

Proof. We will use Sturm’s theorem to prove the claim. Let f j(λ), 0 � j � M be a finite sequence of polynomials with real coefficientssatisfying the conditions: (i) f0(λ) has no roots in some interval [a,b]; (ii) if f j(c) = 0 for c ∈ [a,b] then f j−1(c) f j+1(c) < 0 for 1 � j �M − 1; (iii) if f M(c) = 0 for c ∈ [a,b] then f ′

M(c) f M−1(c) < 0. A sequence satisfying these properties is called a Sturm sequence. Sturm’stheorem states that the number of roots of f M(λ) in [a,b) is equal to the difference χ(b) − χ(a), where χ(c) represents the number ofsign changes of the sequence f0(c), . . . , f M(c).

Z. Rapti / Physics Letters A 377 (2013) 1543–1553 1547

Consider the sequence of polynomials defined in (6) and (7). From our previous results, it follows that it is a Sturm sequence.Indeed, (i) f0(λ) ≡ 1; (ii) f j(c) = 0 implies f j−1(c) f j+1(c) = −a2

j f 2j−1(c) < 0 for 1 � j � M − 1; (iii) f ′

M(c) f M−1(c) = − f 2M−1(c) −

a2M−1( f M−1(c) f ′

M−2(c) − f ′M−1(c) f M−2(c)), and by defining h j(λ) = f j(λ) f ′

j−1(λ) − f ′j(λ) f j−1(λ), it is easy to show that h1(λ) = 1 and

by induction h j(λ) = f 2j−1(λ) + a2

j−1h j−1(λ) > 0 for all λ and 2 � j � M . Then, we apply of Sturm’s theorem on the interval [0,∞) anduse the fact that χ(+∞) = M .

If all oscillators are in-phase, then a j = −1 for all j = 1, . . . , M , from (8) it follows that χ(0) = 0, which gives rise to M positive zerosfor f M . Then, it follows that P M+1 has M + 1 nonnegative zeros that interlace those of f M . Actually, using the same reasoning as above,one can show that there are M − 1 zeros of P M+1 between the M zeros of f M , one zero to the right of the largest zero of f M plus thezero at the origin.

Similarly, if all pairs of adjacent oscillators are out-of-phase, then a j = γ > 0 for all j = 1, . . . , M , hence from (8) it follows thatχ(0) = M , which gives rise to 0 positive eigenvalues for f M . Again, it is easy to see that the zeros of P M+1 interlace those of f M , so thereis the zero at the origin (to the right of the largest zero of f M ), a zero of P M+1 to the left of the smallest zero of f M , and M − 1 negativezeros between the M zeros of f M . �

Therefore, we have given an alternative proof to Theorem 6 in [4] that states that for ε > 0 the time reversible, in-phase multibreathersare unstable with soft onsite potential and the out-of-phase multibreathers are stable with soft onsite potential.

Theorem 2. Multibreathers with in- and out-of-phase oscillators are unstable.

Proof. Consider the general case, where there are M0 positive terms in {σ jσ j+1}Mn=1, which is equivalent to M0 negative a j ’s, which in

turn is equivalent to M0 sign agreements in (8) since f j(0) f j−1(0) > 0 for the relevant j’s. Alternatively, this is equivalent to M − M0 signchanges in (8), which leads to M0 positive zeros for f M and M − M0 negative zeros, by Sturm’s theorem. Arguing as above, P M+1 hasM − M0 negative zeros, one zero at the origin, and M0 positive zeros. �

This proves the conjecture in [4] based on numerical observations, which has been proved by different methods in [6] and [8].

Remark 1. The statements of the previous two theorems hold for general soft potentials. We refer to the soft Morse potential for illustrativepurposes, because it is easier to handle analytically. For hard potentials, the statements of Theorem 1 are reversed, namely the in-phase(out-of-phase) multibreathers are stable (unstable). Theorem 2 is true for any soft or hard onsite potential.

3.2. Second-nearest neighbors

3.2.1. Stability of in-phase and anti-phase multibreathersWe consider the case with interactions with neighbors up to two sites away with Hamiltonian

H =N∑

n=1

1

2y2

n + V (yn) + ε

(a

2(yn − yn−1)

2 + b

2(yn − yn−2)

2)

. (11)

The coupling matrix is a band 5-diagonal matrix and the perturbation matrix for an M-site breather with all oscillators in-phase becomes:

Q =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a + b −a −b 0 0 · · · 0 0 0−a 2a + b −a −b 0 · · · 0 0 0−b −a 2(a + b) −a −b · · · 0 0 0...

......

......

......

...

0 0 0 0 0 · · · 2(a + b) −a −b0 0 0 0 0 · · · −a 2a + b −a0 0 0 0 0 · · · −b −a a + b

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

One can easily check that det Q = 0, hence 0 is an eigenvalue and that by a simple application of Gerschgorin’s theorem (see [22,p. 135]) bounds for the eigenvalues can be obtain. The theorem states that if we set Ri = ∑

i �= j |qij| and denote by Zi the disc inthe complex plane with center qii and radius Ri : Zi = {z ∈ C: |z − qii | � Ri}, then the eigenvalues belong to one of the discs. In ourcase, q11 = a + b = qMM , q22 = 2a + b = qM−1M−1, qnn = 2(a + b), for 3 � n � M − 2 and R1 = |a| + |b| = RM , R2 = 2|a| + |b| = RM−1,Rn = 2(|a| + |b|) for 3 � n � M − 2. Since the perturbation matrix Q is symmetric, we also know that all eigenvalues are real. Therefore,for positive coupling constants a > 0,b > 0 it follows that the eigenvalues belong to the interval [0,4(a + b)], namely all eigenvalues arenonnegative. We only need to verify that the zero eigenvalue is simple, so that the first order perturbation theory holds.

This can be seen by considering the characteristic polynomial P M(λ) = det(Q − λI) = (−1)MλM + · · · + p1Mλ, where the coefficients ofthe first order term are given by p1M = −MrM , and the rM satisfy the recurrence relation

rM − (a + 2b)rM−1 + b2rM−2 = 0,

for M � 4. For the details, see Appendix A.2. The solutions to this recurrence relation can be found in terms of the roots

r± = a + 2b

2± 1

2

√(a + 2b)2 − 4b2 > 0

of the quadratic polynomial

1548 Z. Rapti / Physics Letters A 377 (2013) 1543–1553

r2 − (a + 2b)r + b2 = 0,

using the conditions r2 = a and r3 = a(a + 2b). In fact,

rM = arM−1+ − rM−1−

r+ − r−> 0 for M � 4,

since r+ > r− > 0. Thus, p1M < 0 which shows that the zero eigenvalue is simple.Hence, we have proved the following:

Theorem 3. Multibreathers with all oscillators in-phase are unstable in the case of attracting interactions with the nearest and next-nearest neighbors.

An identical argument yields:

Corollary 1. Multibreathers with all adjacent oscillators out-of-phase are unstable in the case of repulsive interactions with the nearest neighbor andattracting interactions with the next-nearest neighbor.

Proof. In the previous perturbation matrix, if the code is ±[1,−1,1,−1,1,−1, . . .], then q jj+1 = q jj−1 = γ a, q jj+2 = q jj−2 = −b, andq jj±k = 0 for k > 2, which implies that q jj = b − γ a for j = 1, M , q jj = b − 2γ a for j = 2, M − 1, and q jj = 2(b − γ a) for 3 � j � M − 2.Also, R j = |γ a|+ |b|, for j = 1, M , R j = 2|γ a|+ |b|, for j = 2, M −1, and R j = 2(|γ a|+ |b|), for 3 � j � M −2. Since for the Morse potentialγ = ω > 0 and by assumption a < 0 and b > 0, it follows that the results above are true if one replaces −a with γ a. �Remark 2. For any potential, it holds 0 < γ � 1 [4], so Theorem 3 and Corollary 1 are true for any soft potential. For hard potentials, thestability of their respective multibreather configurations is reversed.

Remark 3. If a,b < 0 and the potential is soft, then the in-phase multibreathers are stable. This is true because the discs are now onthe left half-plane, and the eigenvalue at the origin is again simple. This holds since now r− < r+ < 0 and rM is positive for odd M , andnegative for even M . Analogous to Corollary 1 is the case a > 0 and b < 0, which results in stable anti-phase multibreather configurations(for soft potentials).

3.2.2. Phase-shift multibreathersIn [19] multibreather configurations with nontrivial phase profiles were proved to exist analytically and observed numerically. Specif-

ically, through supercritical or subcritical bifurcations these profiles emerge from or collide with the standard ones, where the phasedifferences are 0 or π . The effective Hamiltonian method of MacKay [5] was the theoretical framework used to prove the existence andstability of such multibreathers. The numerical results were generated for an onsite potential of the form V (x) = x2

2 − 0.15 x3

3 − 0.05 x4

4 .In summary, the effective Hamiltonian method consists of averaging over the period of the unperturbed breather and defining properaction-angle variables. The extrema of the effective Hamiltonian determine the relative phases between the adjacent excited oscillators inthe multibreather configuration. The stability can then be investigated by studying the associated Hessian which is related to the Floquetmultipliers of the periodic orbit.

The symmetry coefficient (3) can be modified to account for solutions that are not time-symmetric. Namely, we now define

γnm = −∫ T

2

− T2

y0(t + tn) y0(t + tm)dt

∫ T2

− T2( y0(t))2 dt

. (12)

For the case of the Morse potential this modified coefficient can again be computed analytically and takes the form

γnm = −∑

k k2z2k cos[ωk(tn − tm)]∑

k k2z2k

= ω(1 − ω) − (1 + ω) cos(ω(tn − tm))

(1 + ω2) − (1 − ω2) cos(ω(tn − tm)). (13)

It can easily be seen that for the standard shift tn − tm = T2 we obtain γnm = ω = γ , as expected, and for oscillators in-phase (no shift) it

follows that γnm = −1. This is the only change that occurs in the perturbation matrix and the rest of the theory in [4] remains unaffected.For 3-site multibreathers with coupling constants a = 1 and b = k, the perturbation matrix is

Q =⎛⎝ −γ12 − γ13k γ12 γ13k

γ12 −γ12 − γ23 γ23γ13k γ23 −γ13k − γ23

⎞⎠ ,

with nonzero eigenvalues

λ± = −(γ12 + γ23 + γ13k) ±√

(γ12 + γ23 + γ13k)2 − 3(γ12γ23 + γ23γ13k + γ12γ13k).

In this case, the critical coupling constant kcr which results in a second zero eigenvalue is given by

kcr = − γ12γ23.

γ13(γ12 + γ23)

Z. Rapti / Physics Letters A 377 (2013) 1543–1553 1549

Fig. 1. This figure shows the dependence of the quantity 12kcr

on ω. The results correspond to a Morse onsite potential and a 3-site anti-phase multibreather.

Fig. 2. The figure shows the dependence of t2 − t1 on the critical value kcr . The critical value is defined as the value where a second zero eigenvalue in the perturbationmatrix emerges. The results correspond to a Morse onsite potential and multibreather frequency ω = 0.9.

For the multibreathers with nonstandard phase shifts found in [19] it holds t2 − t1 = t3 − t2 which implies γ12 = γ23. This yieldskcr = − γ12

2γ13. Then, for the anti-phase breather with t2 − t1 = t3 − t2 = T

2 we obtain kcr = ω2 . The critical value kcr of the next-nearest

neighbor coupling depends on the particular onsite potential and on the frequency of the breather. In [19] the critical value kcr wasestimated to be around 0.5 for breathers of low amplitude, which accounts to the breather frequency ω being close to the phononfrequency ω0 (which in their case was 1). For the Morse potential that we are using, ω0 = 1 as well, and for ω → ω0 it holds

kcr → 0.5

which shows that the two results agree. This is expected since near the linear frequency the exact form of the onsite potential doesnot matter. This is true because the coefficient γ → 1 when ω → ω0 = 1, since at this limit, y0(t + T /2) = − y0(t) as in the symmetricpotential case.

Remark. The function f (φ) = 12

∑∞m=1 m2 A2

m cos(mφ) in [19] is the same as the numerator in γnm given in (13). Actually, for the anti-

phase breather φ1 = φ2 = π or equivalently t2 − t1 = t3 − t2 = T2 , it holds kcr = − f (π)

2 f (0)⇒ 1

2kcr= | f (0)

f (π)|, which in turn implies 1

2kcr= 1

γ ( T2 )

and in the case of the Morse potential γ ( T2 ) = ω, so the two expressions for kcr agree. In particular, the dependence of 1

2kcron ω is shown

in Fig. 1, where we plot 12kcr = 1

ω . The plot is qualitatively similar to the one obtained by numerical calculations in [19].In Fig. 2 we plot the critical value kcr as a function of the time-shift t2 − t1 = t3 − t2 of adjacent oscillators for ω = 0.9. This quantity is

given by kcr = − γ122γ13

, with t2 − t1 as the independent variable. We also use the fact that t3 − t1 = t3 − t2 + t2 − t1 = 2(t2 − t1) to evaluateγ12 and γ13 as a function of the time-shift t2 − t1 only. As a reference, we also plot the line corresponding to the standard time-shiftt2 − t1 = T

2 . We must point out that our figure is based on the stability analysis alone, assuming that multibreathers with such phaseshifts exist. The range of time shifts was chosen based on the numerical observations in [19].

We also consider a 4-site multibreather with t2 − t1 = t3 − t2 = t4 − t3 = T2 which implies γ12 = γ23 = γ34 = ω and γ13 = γ24 = −1.

The perturbation matrix now becomes

1550 Z. Rapti / Physics Letters A 377 (2013) 1543–1553

Q =

⎛⎜⎜⎜⎝

−γ12 − γ13k γ12 γ13k 0

γ12 −γ12 − γ23 − γ24k γ23 γ24k

γ13k γ23 −γ13k − γ23 − γ34 γ34

0 γ24k γ34 −γ24k − γ34

⎞⎟⎟⎟⎠ ,

=

⎛⎜⎜⎝

k − ω ω −k 0ω k − 2ω ω −k−k ω k − 2ω ω0 −k ω k − ω

⎞⎟⎟⎠

with nonzero eigenvalues

λ = 2(k − ω) and λ± = k − 2ω ±√

(k − 2ω)2 − 2ω(ω − 3k).

Therefore, the critical values

kcr = ω,ω

3,

are the same as the ones reported in [19] for ω → ω0.

3.3. Third-nearest neighbors

We consider the case with interactions with neighbors up to three sites away with Hamiltonian

H =N∑

n=1

1

2y2

n + V (yn) + ε

(a

2(yn − yn−1)

2 + b

2(yn − yn−2)

2 + c

2(yn − yn−3)

2)

. (14)

The coupling matrix is a band 7-diagonal matrix and although the algebraic approach of the previous section can be applied, the calcu-lations required to study the stability of M-site multibreathers are too involved. Instead, we will focus on 4-site multibreathers and theirstability properties as the coupling parameters vary.

Specifically, we will assume that the coupling constants a = 1 and c = k3 are fixed, while b = k2 will be considered as a variable. Theperturbation matrix for t2 − t1 = t3 − t2 = t4 − t3 = T

2 which implies that γ12 = γ23 = γ34 = ω = γ14 and γ13 = γ24 = −1 is

Q =

⎛⎜⎜⎜⎝

−γ12 − γ13k2 − γ14k3 γ12 γ13k2 γ14k3

γ12 −γ12 − γ23 − γ24k2 γ23 γ24k2

γ13k2 γ23 −γ13k2 − γ23 − γ34 γ34

γ14k3 γ24k2 γ34 −γ14k3 − γ24k2 − γ34

⎞⎟⎟⎟⎠ ,

=

⎛⎜⎜⎝

k2 − ω − ωk3 ω −k2 ωk3ω k2 − 2ω ω −k2

−k2 ω k2 − 2ω ωωk3 −k2 ω k2 − ω − ωk3

⎞⎟⎟⎠

with nonzero eigenvalues

λ = 2(k2 − ω) and λ± = k2 − 2ω − ωk3 ±√

(k2 − 2ω − ωk3)2 − 2ω(3ωk3 + ω − k2(k3 + 3)

).

Therefore, the critical values of k2 as a function of ω and k3 are

k2,cr = ω,ω1 + 3k3

k3 + 3. (15)

This verifies the numerical observation in [19] that the critical value of k2, where the stability of the multibreather is lost (and wherethe supercritical pitchfork bifurcation occurs) depends strongly on k3, while the critical value where the stability of the multibreather isregained (and where the subcritical pitchfork bifurcation occurs) does not depends on it. We also note that if k3 = 0 we retrieve the criticalvalue ω

3 from the case with next-nearest interaction only. We plot the critical values given in (15) for ω = 0.9 in Fig. 3. The dependenceon k3 is qualitatively the same as the one in the analogous figure in [19]. The reason for this agreement is because at the locations of thebifurcation, the stability of the out-of-phase multibreather changes, hence our stability analysis suffices to capture the critical values.

4. Discussion and conclusions

We have presented results on multibreather stability in Klein–Gordon chains with interactions including the nearest, next-nearest,and third-nearest neighbor. We have provided an alternative proof for the instability of multibreathers with all oscillators in-phase andnearest neighbor interaction only. Also, we demonstrated, in a more algebraic way, that the number of positive/negative eigenvalues of anappropriately defined perturbation matrix is related to the number of adjacent oscillators that are in/out-of phase in the multibreatherconfiguration. This result was also obtained in [6] with a different method. We then showed how these results can be extended to thecase of interactions beyond the nearest neighbor.

Z. Rapti / Physics Letters A 377 (2013) 1543–1553 1551

Fig. 3. The figure shows the dependence of the critical value of k2 on k3. The critical value is defined as the value where a second zero eigenvalue in the perturbation matrixemerges. The results correspond to a Morse onsite potential and multibreather frequency ω = 0.9.

We have also extended the method of Archilla [4] to study analytically the properties and bifurcations of multibreathers that havenonstandard shifts such as those recently reported in [19]. Our analytical results agree with the ones obtained in [19] and this suggeststhat this method might be used to make predictions on the locations on the sub- and supercritical pitchfork bifurcations that occur as thecoupling parameters vary.

Acknowledgements

We gratefully acknowledge fruitful discussions with P.G. Kevrekidis and D. Pelinovsky, as well as the support of NSF-DMS-0708421.

Appendix A

A.1. Simple zero root in the nearest-neighbor case

P ′M+1(0) = − f M(0) − aM

(f ′

M(0) + aM f ′M−1(0)

)= − f M(0) − aM XM , with X j = f ′

j(0) + a j f ′j−1(0).

It also holds that

f ′j(0) = − f j−1(0) − (a j + a j−1) f ′

j−1(0) − a2j−1 f ′

j−2(0), j = 2, . . . , M,

with

f ′1(0) = −1, f ′

0(0) = 0.

Hence, for 2 � j � M it follows that

X j = − f j−1(0) − a j−1 X j−1,

which yields

X j = (−1) j−2a2 · · ·a j−1 X2 +j−2∑k=1

(−1)ka j−1 · · ·a j−k+1 f j−k(0)

= 2(−1) ja1 · · ·a j−1 +j−2∑k=1

(−1) ja1 · · ·a j−1

= j(−1) ja1 · · ·a j−1.

Then,

P ′M+1(0) = −(−1)Ma1 · · ·aM − aM M(−1)Ma1 · · ·aM−1 = (−1)M+1(M + 1)a1 · · ·aM .

1552 Z. Rapti / Physics Letters A 377 (2013) 1543–1553

A.2. Next-nearest neighbor interaction determinant

In the expression for the determinant of Q − λI by adding all columns to the first one, once notices that −λ is a common factor.Hence, P M(λ) = −λqM(λ), which implies P ′

M(0) = −qM(0), where

qM(0) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 −a −b 0 0 · · · 0 0 0 01 2a + b −a −b 0 · · · 0 0 0 01 −a 2(a + b) −a −b · · · 0 0 0 01 −b −a 2(a + b) −a · · · 0 0 0 0...

......

......

......

......

1 0 0 0 0 · · · −a 2(a + b) −a −b1 0 0 0 0 · · · −b −a 2a + b −a1 0 0 0 0 · · · 0 −b −a a + b

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

Next, by adding all rows to the first one, we obtain

qM(0) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

M 0 0 0 0 · · · 0 0 0 01 2a + b −a −b 0 · · · 0 0 0 01 −a 2(a + b) −a −b · · · 0 0 0 01 −b −a 2(a + b) −a · · · 0 0 0 0...

......

......

......

......

1 0 0 0 0 · · · −a 2(a + b) −a −b1 0 0 0 0 · · · −b −a 2a + b −a1 0 0 0 0 · · · 0 −b −a a + b

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣or equivalently

qM(0) = M

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

2a + b −a −b 0 · · · 0 0 0 0−a 2(a + b) −a −b · · · 0 0 0 0−b −a 2(a + b) −a · · · 0 0 0 00 −b −a 2(a + b) · · · 0 0 0 0...

......

......

......

...

0 0 0 0 · · · −a 2(a + b) −a −b0 0 0 0 · · · −b −a 2a + b −a0 0 0 0 · · · 0 −b −a a + b

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

Adding all rows to the first and then all columns to the first one, we finally obtain

qM(0) = M

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

a + b b 0 0 · · · 0 0 0b 2(a + b) −a −b · · · 0 0 00 −a 2(a + b) −a · · · 0 0 00 −b −a 2(a + b) · · · 0 0 0...

......

......

......

...

0 0 0 0 · · · 2(a + b) −a −b0 0 0 0 · · · −a 2a + b −a0 0 0 0 · · · −b −a a + b

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= MrM .

It now follows that

rM = (a + b)βM − b2βM−1,

where

βM =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

a + 2b b 0 0 · · · 0 0 0b 2(a + b) −a −b · · · 0 0 00 −a 2(a + b) −a · · · 0 0 00 −b −a 2(a + b) · · · 0 0 0...

......

......

......

...

0 0 0 0 · · · 2(a + b) −a −b0 0 0 0 · · · −a 2a + b −a0 0 0 0 · · · −b −a a + b

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣satisfies the recurrence relation

βM = (a + 2b)βM−1 − b2βM−2.

Z. Rapti / Physics Letters A 377 (2013) 1543–1553 1553

From these two recurrence relations we obtain

rM = aβM + b(βM − bβM−1) and βM+1 = (a + b)βM + b(βM − bβM−1)

which implies that

rM = βM+1 − bβM and aβM = rM − brM−1.

Finally multiplying the first with a and using the second one, yields

rM+1 − (a + 2b)rM + b2rM−1 = 0.

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