Parametrically Excited Hamiltonian Partial Differential Equations

Abstract

Consider a linear autonomous Hamiltonian system with a time periodic bound state solution. In this paper we study the structural instability of this bound state ^M relative to time almost periodic perturbations which are small, localized and Hamiltonian. This class of perturbations includes those whose time dependence is periodic, but encompasses a large class of those with finite (quasiperiodic) or infinitely many non-commensurate frequencies. Problems of the type considered arise in many areas of application including ionization physics and the propagation of light in optical fibers in the presence of defects. The mechanism of instability is radiation damping due to resonant coupling of the bound state to the continuum modes by the time-dependent perturbation. This results in a transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. These results generalize those of A. Soffer and M.I. Weinstein, who treated localized time-periodic perturbations of a particular form. In the present work, new analytical issues need to be addressed in view of (i) the presence of infinitely many frequencies which may resonate with the continuum as well as (ii) the possible accumulation of such resonances in the continuous spectrum. The theory is applied to a general class of Schrödinger operators.