1 Introduction.- 1.1 Invariant Manifolds in Infinite Dimensions.- 1.2 Aims and Scope of This Monograph.- 2 The Perturbed Nonlinear Schrödinger Equation.- 2.1 The Setting for the Perturbed Nonlinear Schrödinger Equation.- 2.2 Spatially Independent Solutions: An Invariant Plane.- 2.3 Statement of the Persistence and Fiber Theorems.- 2.4 Explicit Representations for Invariant Manifolds and Fibers.- 2.5 Coordinates Centered on the Resonance Circle.- 2.5.1 Definition of the H Norms.- 2.5.2 A Neighborhood of the Circle of Fixed Points.- 2.5.3 An Enlarged Phase Space.- 2.5.4 Scales Through 6.- 2.5.5 The Equations in Their Final Setting.- 2.6 (6 = 0) Invariant Manifolds and the Introduction of a Bump Function.- 2.6.1 (6 = 0) Invariant Manifolds.- 2.6.2 Tangent and Transversal Bundles of M.- 2.6.3 Introduction of a Bump Function.- 2.6.4 Existence, Smoothness, and Growth Rates for the "Bumped" Flow in the Enlarged Phase Space.- 3 Persistent Invariant Manifolds.- 3.1 Statement of the Persistence Theorem and the Strategy of Proof.- 3.2 Proof of the Persistence Theorems.- 3.2.1 Definition of the Graph Transform.- 3.2.2 The Graph Transform as a C° Contraction.- 3.3 The Existence of the Invariant Manifolds.- 3.4 Smoothness of the Invariant Manifolds.- 3.5 Completion of the Proof of the Proposition.- 4 Fibrations of the Persistent Invariant Manifolds.- 4.1 Statement of the Fiber Theorem and the Strategy of Proof.- 4.2 Rate Lemmas.- 4.3 The Existence of an Invariant Subbundle E.- 4.4 Smoothness of the Invariant Subbundle E.- 4.5 Existence of Fibers.- 4.6 Smoothness of the Fiber fE(Q) as a Submanifold.- 4.7 Metric Characterization of the Fibers.- 4.8 Smoothness of Fibers with Respect to the Base Point.- References.
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, as well as for the existence and persistence of fibrations of these invariant manifolds. Their techniques are based on an infinite dimensional generalisation of the graph transform and can be viewed as an infinite dimensional generalisation of Fenichels results. As such, they may be applied to a broad class of infinite dimensional dynamical systems.
- Presents detailed and pedagogic proofs - The authors techniques can be applied to a broad class of infinite dimensional dynamical systems - Stephen Wiggins has authored many successful Springer titles and is the editor of Springers Journal of Nonlinear Science, currently the number one cited journal in applied mathematics