Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations

ISBN-13: 9781461273073 / Angielski / Miękka / 2012 / 172 str.

Charles Li; Stephen Wiggins

Opis

This book presents a development of invariant manifold theory for a spe- cific canonical nonlinear wave system -the perturbed nonlinear Schrooinger equation. The main results fall into two parts. The first part is concerned with the persistence and smoothness of locally invariant manifolds. The sec- ond part is concerned with fibrations of the stable and unstable manifolds of inflowing and overflowing invariant manifolds. The central technique for proving these results is Hadamard's graph transform method generalized to an infinite-dimensional setting. However, our setting is somewhat different than other approaches to infinite dimensional invariant manifolds since for conservative wave equations many of the interesting invariant manifolds are infinite dimensional and noncom pact. The style of the book is that of providing very detailed proofs of theorems for a specific infinite dimensional dynamical system-the perturbed nonlinear Schrodinger equation. The book is organized as follows. Chapter one gives an introduction which surveys the state of the art of invariant manifold theory for infinite dimensional dynamical systems. Chapter two develops the general setup for the perturbed nonlinear Schrodinger equation. Chapter three gives the proofs of the main results on persistence and smoothness of invariant man- ifolds. Chapter four gives the proofs of the main results on persistence and smoothness of fibrations of invariant manifolds. This book is an outgrowth of our work over the past nine years concerning homoclinic chaos in the perturbed nonlinear Schrodinger equation. The theorems in this book provide key building blocks for much of that work.

Spis treści

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.