Non-existence of phase-shift breathers in one-dimensional Klein–Gordon lattices with nearest-neighbor interactions
Introduction
Since [21], [17] much interest has been drawn in the study of space-localized and time-periodic motions in lattices of coupled oscillators. These motions are called discrete breathers (DB) if the oscillation is localized around one “central” lattice site, while, if there are more than one central oscillators, the motion is called multibreather (MB) or multi-site breather. The wide interest about discrete breathers–multibreathers is underlined by the numerous review papers there exist on this subject (e.g. [9], [14], [8], [4]).
One of the most popular systems in which such motions are studied is the well-known Klein–Gordon (KG) chain. The classical KG setting consists of a one-dimensional lattice of oscillators each coupled with its nearest-neighbors (NN). Since the first proof of existence of DBs [15], there have been several papers dealing with the issue of existence and stability of MBs in KG chains (e.g. [1], [20], [7]). In [13] a methodology for proving the existence of multi-site breathers was introduced based in the work of [2], [16] and using also the terminology of [10]. This methodology was generalized for a generic Klein–Gordon chain in [11] and provided general persistence and stability conditions independently of the precise form of the on-site potential. The stability results of [11], have been shown in [6] to be in correspondence to the results of [3], which were already been obtained by using the band theory of [4]. In a recent work [19] an alternative proof of the stability theorem of [3] has been presented by using the same band theory, providing also a proof of the eigenvalue counting result of [3]. These results have been generalized in [18] by also considering “holes” between the central oscillators. In a different context, similar results have been recently obtained [22] by considering a diatomic FPU chain.
The existing multibreather solutions are categorized in terms of the phase differences between the central oscillators. It is well known that KG chains can support multibreathers with phase differences between the successive central oscillators . These are the standard configurations. Although there is strong evidence that phase-shift breathers i.e. multibreathers with phase differences , cannot exist, a rigorous proof of this fact had not been presented.
In this work we prove that the one-dimensional Klein–Gordon lattice with nearest-neighbor interactions cannot support phase-shift breathers, by proving that the persistence conditions provided by [11] do not have solutions other than the standard ones , π.
The fact of the non-existence of phase-shift breathers in KG chains determines also the stability of the standard configurations. In particular, in [11] the main theorem has been stated under the assumption of non-existence of phase-shift configurations. On the other hand, if we consider a KG chain with interactions between its oscillators further than mere the nearest-neighbors, phase-shift breathers can be supported [12] and consequently the stability picture radically changes.
The Letter is organized as follows: in Section 2 we present briefly the methodology for the derivation of the persistence conditions for the existence of multibreathers in KG chains developed in [11], while we introduce some terminology. In Section 3 the main theorem about the non-existence of phase-shift breathers is proven. In Section 4 we discuss the implication of this theorem to the stability of the standard MB configurations.
Section snippets
Persistence and stability of multibreathers in 1D Klein–Gordon lattices with nearest-neighbor interactions
In this section we will shortly present the main results of [11], concerning the existence of multibreathers (MB) in a Klein–Gordon (KG) chain. The classical KG setting is defined as a 1D lattice of coupled oscillators each one moving in a nonlinear potential possessing a local minimum at (, ). Each oscillator is coupled with its two nearest neighbors (NN) with a linear coupling force through a coupling constant ε, as shown in Fig. 1.
The Hamiltonian of a Klein–Gordon
Proof of non-existence of phase-shift breathers in one-dimensional Klein–Gordon chains with nearest-neighbor interactions
In this section we will prove that phase-shift breathers1 ( or π) cannot be supported in one-dimensional Klein–Gordon chains with nearest-neighbor interactions. This fact will be proven by showing that the persistence conditions
Discussion about the stability of multibreathers in Klein–Gordon chains
In [11], under the assumption of non-existence of phase-shift breathers, the spectral stability of the multibreather solutions in one-dimensional Klein–Gordon chains with nearest-neighbor interactions, is well established by the theorem: Theorem 2 Under the assumption that (4) has only the solutions, in systems of the form (1), if the only configuration which leads to linearly stable multibreathers, for small enough, is the one with (anti-phase multibreather), while if
Conclusions
It is well known that one-dimensional Klein–Gordon (KG) lattices with nearest-neighbor (NN) interactions support multibreathers with the standard phase-difference between adjacent central oscillators. On the other hand, there were strong evidences (including numerical computations) suggesting that phase-shift breathers i.e. multibreathers with cannot exist in this classical KG setting.
In the present work we prove that, indeed, the only configurations that can exist in a
Acknowledgements
This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program “THALES. Investing in knowledge society through the European Social Fund”.
The author would like to thank Prof. D.E. Pelinovsky for a conversation which rekindled the interest about this work and Prof. P.G. Kevrekidis for his invaluable help
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