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Metastability of breather modes of time-dependent potentials. (English) Zbl 0979.37038

The authors study solutions of a linear spatially one-dimensional Schrödinger equation (SE) with a potential being time periodic and spatially decaying. It is supposed that the potential is a small perturbation of that being separable (or integrable), i.e., the initial-value problem for the related SE can be solved exactly, such time periodic and spatially decaying potentials were studied by P. D. Miller and N. N. Akhmediev [Physica D 123, No. 1-4, 513-524 (1998; Zbl 0939.35157)] using soliton theory of cubic nonlinear Schrödinger equation. Exact breather modes exist for such potentials. It was found that under the perturbation there are close solutions being metastable, that is, long-lived but eventually decay. The results obtained are used to analyse the frequency detuning in a plane optical waveguide.

MSC:

37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
35C20 Asymptotic expansions of solutions to PDEs
78A60 Lasers, masers, optical bistability, nonlinear optics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
35C15 Integral representations of solutions to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)

Citations:

Zbl 0939.35157
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