On the stability of multibreathers in Klein–Gordon chains

Abstract

Abstract. In the present paper, a theorem, which determines the linear stability of multibreathers excited over adjacent coupled oscillators in Klein-Gordon chains, is proven. Specifically, it is shown that for soft nonlinearities, and positive nearestneighbor inter-site coupling, only structures with adjacent sites excited out-of-phase may be stable, while only in-phase ones may be stable for negative coupling. The situation is reversed for hard nonlinearities. This method can be applied to n-site breathers, where n is any finite number and provides a detailed count of the number of real and imaginary characteristic exponents of the breather, based on its configuration. In addition, an O( √ ε) estimation of these exponents can be extracted through this procedure. To complement the analysis, we perform numerical simulations and establish that the results are in excellent agreement with the theoretical predictions, at least for small values of the coupling constant ε.

An extensive resonant normal form for an arbitrary large Klein–Gordon model

“…We close the comments on the GdNLS with the remark that it could be the starting point for several applications which concern the Klein-Gordon model, like the variational approximations for breather solutions (see, e.g., [7,8]), the approximation of the small amplitude Cauchy problem, the existence and linear stability of multibreathers (see, e.g., [15,16,20,21]). Moreover there are several recent works on models with more than first neighbor interactions or with different nonlinearities, like [6,17,23,26] where spatially localized periodic orbits, as breathers or multibreathers, are studied: with respect to this, we expect that the approach proposed in the present paper can be suitably extended in these more general models, leading to different GdNLS-like normal forms.…”

Multibreathers in Klein–Gordon chains with interactions beyond nearest neighbors

“…Since the action J i remains constant along an orbit in the anticontinuum limit, x i depends only on w i . So, the average value of H 1 becomes ( [22] appendix A)…”

Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

“…In particular, exploring the limit of weak coupling between the nonlinear oscillators, existence [26] and stability [2,4] of the fundamental (single-site) breathers were established (see also the recent works in [30,31]). More complicated multi-breathers were classified from the point of their spectral stability in the recent works [1,25,33]. Nonlinear stability and instability of multi-site breathers were recently studied in [13].…”