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A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations. (English) Zbl 0969.35123

Authors’ abstract: A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schrödinger type equation. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
78A50 Antennas, waveguides in optics and electromagnetic theory
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
47D03 Groups and semigroups of linear operators
47D06 One-parameter semigroups and linear evolution equations
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