Resonances in parametrically excited Hamiltonian partial differential equations.

Abstract

In this thesis, we consider a linear autonomous Hamiltonian system with finitely many, say <italic>m</italic>, time periodic bound state solutions. We study their dynamics under time dependent perturbations which are small, localized in space and Hamiltonian. The time evolution of the perturbations ranges from almost periodic to trains of short lived pulses. The analysis proceeds through a reduction of the original infinite dimensional dynamical system to the dynamics of two coupled subsystems: a dominant <italic> m</italic>-dimensional system of ordinary differential equations (<italic> normal form</italic>), governing the projections onto the bound states and an infinite dimensional dispersive wave equation. Compared to the existing literature, the interaction picture is considerably more complicated and requires deeper analysis, due to a multiplicity of bound states and the very general nature of the perturbation's time dependence. Parametric forcing induces coupling of bound states to continuum radiation modes, bound states directly to bound states, as well as coupling among bound states, which is mediated by continuum modes. Our analysis elucidates these interactions and we derive an explicit dominant evolution on long time scales. We prove the metastability (long life time) and eventual decay of bound states for a large class of systems. We also show that certain trains of pulse like perturbation induce diffusion of energy among the bound states on the time scales on which the latter are metastable. Problems of the type considered arise in many areas of application including ionization physics, quantum molecular theory and the propagation of light in optical wave guides in the presence of defects.