Mathematics – Dynamical Systems
Scientific paper
2000-02-10
Mathematics
Dynamical Systems
69 pages, 13 figures, to appear in Nonlinearity
Scientific paper
10.1088/0951-7715/13/3/303
We study the solutions of linear Schroedinger 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 Schroedinger equation solved by Miller and Akhmediev. 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 $\epsilon^{-2}$, 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 $\epsilon^{-1}$ 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 and extended by Kirr and Weinstein to justify the expansions and also 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.
Miller Peter D.
Soffer Abner
Weinstein Michael I.
No associations
LandOfFree
Metastability of Breather Modes of Time-Dependent Potentials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Metastability of Breather Modes of Time-Dependent Potentials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Metastability of Breather Modes of Time-Dependent Potentials will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-446125