Decay of coherent structures in damped Hamiltonian systems

Abstract

In numerical and physical experiments it is often observed that finite and infinite dimensional systems exhibit a low dimensional behaviour, in the sense that the dynamics looks as if it can be described with a few parameters. Often a spatially coherent structure is characteristic for the phenomenon. A well known example is a solitary wave, characterized by its phase and its amplitude. In this paper we consider a Hamiltonian system (or a Poisson system), that has an additional constant of motion (besides the Hamiltonian). We show coherent structures in such a system, by describing some solutions with 2 parameters, induced by the constant of motion. Further we demonstrate that the coherent structures survive even in cases where a small perturbation, such as dissipation, is present. This is demonstrated in some detail for a spherical pendulum with uniform friction, for the Korteweg-de Vries equation with uniform damping and for the Korteweg-de Vries-Burgers equation.