Dispersion and dissipation: a model problem

Structural stability and the persistence of coherent structures under perturbations of a dynamical system are fundamental issues in dynamical systems theory with implications in many fields of application. In the context of discrete, finite dimensional Hamiltonian systems, this issue is addressed by the celebrated KAM theorem, which guarantees the persistence of most invariant tori of the unperturbed dynamics under small Hamiltonian perturbations. For infinite dimensional systems, defined by Hamiltonian partial di↵erential equations (PDEs), KAM type methods have recently been used to obtain results on the persistence of periodic and quasi-periodic solutions, in the case where solutions are defined on compact spatial domains with appropriate boundary conditions. The compactness of the spatial domain ensures discreteness of the spectrum associated with the unperturbed dynamics. Therefore this situation is the generalization of the finite dimensional case to systems with an infinite number of discrete oscillators and frequencies. In this notes we consider these questions in the context of Hamiltonian systems for which the unperturbed dynamics has associated with it discrete and continuous spectrum. This situation arises in the study of Hamiltonian PDEs governing functions defined on unbounded spatial domains or, more generally, extended systems. The physical picture is that of a system which can be viewed as an interaction between one or more discrete oscillators and a field or continuous medium. In contrast to the KAM theory, where nonresonance implies persistence, we find here that resonant nonlinear interaction between discrete (bound state) modes and continuum (dispersive radiation) modes leads to energy transfer from the discrete to continuum modes. This mechanism is responsible for the eventual time-decay and nonpersistence of trapped states. The rate of timedecay, however, is very slow and hence such a trapped state can be thought of as a metastable state. This paper is devoted to the study of the following questions: