Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing

Abstract

Hamiltonian classical systems are the farthest from variational systems. Formally they cannot decay because the equation of motion is reversible. Thus the long time behavior of their solutions is naturally described by statistical methods. For instance, a large assembly of interacting particles in a box follow the Gibbs-Boltzmann statistics, as far as average properties are concerned. As noticed already by Jeans, this statistical approach does not hold for the long time behavior of classical nonlinear field equations, because of the infinite heat capacity of a classical field. Thus nonlinear and nonintegrable classical fields cannot relax for long times toward an ill defined thermal equilibrium. The specific example considered here is the long time dynamics of solutions of the nonlinear Schrödinger equation, in the defocusing case. I show, under some assumptions, that for long times a cascade of energy toward smaller and smaller scales introduces a kind of dissipation in a system that is formally reversible, and I give the scaling laws for this. Besides this relaxation by weak interaction between phonons, there is also a relaxation of localized vortical structures emitting phonons.