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Metastability of Breather Modes of Time Dependent Potentials

P. D. Miller
Institute for Advanced Study, Princeton
Department of Mathematics and Statistics
Monash University, Clayton, Vic 3168 Australia

A. Soffer
Department of Mathematics
Rutgers University
Piscataway, New Jersey 08854-8019

M. I. Weinstein
Mathematical Sciences Research
Bell Labs 2C-358, Lucent Technologies
600 Mountain Avenue
Murray Hill, New Jersey, 07974
Department of Mathematics
University of Michigan


We study the solutions of linear Schrödinger equations in which the potential energy is a periodic function of time and is sufficiently localized in space. We consider the potential to be close to one that is time periodic and yet explicitly solvable. A large family of such potentials has been constructed and the corresponding Schrödinger equation solved by Miller and Akhmediev [MA98]. Exact bound states, or breather modes, exist in the unperturbed problem and are found to be generically metastable in the presence of small periodic perturbations. Thus, these states are long-lived but eventually decay. On a time scale of order one over epsilon squared, where epsilon is a measure of the perturbation size, the decay is exponential, with a rate of decay given by an analogue of Fermi's golden rule. For times of order one over epsilon the breather modes are frequency shifted. This behavior is derived first by classical multiple scale expansions, and then in certain circumstances we are able to apply the rigorous theory developed by Soffer and Weinstein [SW98b] and extended by Kirr and Weinstein [KW99a] to justify the expansions and also to provide longer time asymptotics that indicate eventual dispersive decay of the bound states with behavior that is algebraic in time. As an application, we use our techniques to study the frequency dependence of the guidance properties of certain optical waveguides. We supplement our results with numerical experiments.