Stability of discrete breathers in nonlinear Klein-Gordon type lattices with pure anharmonic couplings

Abstract

We consider the discrete breathers in one-dimensional nonlinear Klein-Gordon type lattices with pure anharmonic couplings. A discrete breather in the limit of vanishing couplings, i.e., the anti-continuous limit, consists of a number of in-phase or anti-phase excited particles, separated by particles at rest. Existence of the discrete breathers is proved for weak couplings by continuation from the anti-continuous limit. We prove a theorem which determines the linear stability of the discrete breathers. The theorem shows that the stability or instability of a discrete breather depends on the phase difference and distance between the two sites in each pair of adjacent excited sites in the anti-continuous solution. It is shown that there are two types of the dependence determined by the sign of αɛ, where α and ɛ are parameters such that positive (respectively, negative) α represents hard (respectively, soft) on-site nonlinearity and positive (respectively, negative) ɛ represents attractive (respectively, repulsive) couplings.