A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schrödinger Equations

Details A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schrödinger Equations A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schrödinger Equations

1999-07-17

arxiv.org/abs/solv-int/9907017v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

LaTeX, 16 pages

Scientific paper

A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schr\" odinger type equations. 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.

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