Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations
By: by Charles Li, Stephen Wiggins
Contributors: Wiggins, Stephen
Series: Applied Mathematical Sciences -- 128; Applied mathematical sciences (Springer-Verlag New York Inc.) -- 128.
Publisher: Springer New York
Published: 1997
Place of publication: New York, NY
Language: eng
Format: [electronic resource] /
Number of pages: 172
ISBN-13: 9781461273073; 9781461218388
Subjects: Geometry; Mathematics; Cell aggregation; Global analysis (Mathematics)
Description: The nonlinear Schroedinger (NLS) equation is a fundamental nonlinear partial differential equation (PDE) that arises in many areas and engineering, e.g. in plasma physics, nonlinear waves, and nonlinear optics. It is an example of a completely integrable PDE where phase space structure is known in some detail. In this monograph the authors present detailed and pedagogic proofs of persistence theorems for normally hyperbolic invariant manifolds and their stable and unstable manifolds for classes of perturbations of the NLS equation. The existence and persistence of fibrations of these invariant manifolds is also proved. The authors' techniques are based on an infinite dimensional generalization of the graph transform and can be viewed as an infinite dimensional generalization of Fenichel's results. This book also shows that the authors' techniques are quite general and can be applied to a broad class of infinite dimensional dynamical systems.
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Table of contents: title: Introduction; title: Invariant Manifolds in Infinite Dimensions. Aims and Scopes of This Monograph; title: The Perturbed Nonlinear Schroedinger Equation. The Setting for the Perturbed Nonlinear Schroedinger Equation. Spatially Independent Solutions: An Invariant Plane. Statement of the Persistence and Fiber Theorems. Explicit Representations for Invariant Manifolds and Fibers. Coordinates Centered on the Resonance Circle. Definition of the H Norms. A Neighborhood of the Circle of Fixed Points. An Enlarged Phase Space. Scales through _. The Equations in their Final Setting. (_=0) Invariant Manifolds and the Introduction of a Bump Function. (_=0) Invariant Manifolds.