Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations

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.