Numerical computation of travelling breathers in Klein–Gordon chains

Abstract

We numerically study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions having the property of being exactly translated by p sites along the chain after a fixed propagation time T (these solutions generalize the concept of solitary waves for which p = 1). In the case of even on-site potentials, the existence of small amplitude travelling breathers superposed on a small oscillatory tail has been proved recently (G. James and Y. Sire, to appear in Comm. Math. Phys., 2004), the tail being exponentially small with respect to the central oscillation size. In this paper we compute these solutions numerically and continue them into the large amplitude regime for different types of even potentials. We find that Klein-Gordon chains can support highly localized travelling breather solutions superposed on an oscillatory tail. We provide examples where the tail can be made very small and is difficult to detect at the scale of central oscillations. In addition we numerically observe the existence of these solutions in the case of non even potentials.

Discrete breathers in two-dimensional anisotropic nonlinear Schrödinger lattices

“…Also in 2.1, we discuss briefly the most relevant formal differences with respect to alternative approaches to (onedimensional) exact mobility of discrete breathers, e.g. those in refs [29] and [30,31]. In subsection 2.2 we introduce the two-dimensional anisotropic Salerno lattice and provide explanations on the implementation of the numerical procedures used to study the dynamics of 2D discrete breathers.…”

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

“…Results are taken from reference [24]. In the normal form (6.40), coefficient s2 (determined by expression (6.43)) is negative for β > 0 with (T0, γ0) lying on the left side of a curve Γm, and for β < 0 with (T0, γ0) lying on the right side.…”

“…The center manifold reduction provides a leading order approximation of bifurcating homoclinics, which is used as an initial guess. We refer the reader to [24] for further details on this numerical method. Another technique which can be used for the computation of solitary waves exploits a modulational instability [24,57].…”

“…Pulsating traveling waves of this type have been numerically computed by the present authors in [24], and [25] treats the case of traveling waves. The existence of traveling kinks showing similar oscillatory tails is reported in several references (see, e.g.…”

“…6.3.1 and 6.4.2), based on center manifold theory, which examine the existence of breathers and nanopterons in FPU and KG lattices respectively. Numerical computations from references [24,34] are also presented to illustrate the mathematical results. On the other hand we give an overview of the general center manifold theorems which have been applied in this context, considering both infinitedimensional differential equations [35,36] and quasilinear maps [28].…”