Chaotic Behavior Induced by Spatially Inhomogeneous Structures such as Solitons

Abstract

One reason for the recent increasing interest in chaotic systems is due to the simplicity of the models: even a simple deterministic system with a few degrees of freedom shows chaotic behavior. Simple models with chaotic behavior are often used as models of systems with many degrees of freedom, when only small numbers of them are relevant to the evolution. This modeling is powerful for the investigation near the onset of the chaos. After one knows that very simple systems show chaotic behavior, it seems not so surprising if most of nonlinear systems with infinite degrees of freedom show chaotic behavior. Apparent exceptions are found in completely integrable systems. In those systems, time evolution is completely described by nonlinear normal modes including solitons. So far the onset of chaos has mainly been investigated by the reduction of system variables to a few relevant modes. Another approach is, however, possible by starting with the small deviation from the complete integrability when perturbation is added. The onset of chaos can be investigated without the reduction of variables in this case. The purpose of this paper is to show an aspect of the chaotic behavior in perturbed integrable systems. In this paper we restrict ourselves to perturbed sine-Gordon systems. We will show an example where spatial inhomogeneities like solitons have an important contribution to the chaotic behavior [1].