Integrability and localized excitations in nonlinear discrete systems

Abstract

We analyze the origin and features of localized excitations in nonlinear discrete Klein-Gordon systems. We connect the presence of stationary excitations with the existence of local integrability of the original N-degree-of-freedom system. The method consists of constructing a reduced problem of a few degrees of freedom and analyzing its phase-space structure with the help of geometrical methods (Poincar\'e maps). We find a correspondence between regular and chaotic motion in the reduced problem on one side and localized and delocalized states in the infinite systems on the other side. The periodic trajectories corresponding to elliptic fixed points of the Poincar\'e map are related to previous numerical and analytical studies. We analyze the stability of the periodic orbits with respect to small-amplitude phonons as well as the internal stability of multiple-frequency localized excitations. We find an energy threshold for the existence of stationary localized excitations and an energy threshold for the existence of instabilities due to internal resonances (onset of chaos). Approximation schemes accounting for the main properties of stationary localized excitations are applied.