Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations

Authors: Charles Li, Stephen Wiggins

Publisher: Springer New York

Book Series : Applied Mathematical Sciences

Included in: Professional Book Archive

About this book

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.

Frontmatter

1. Introduction

Abstract

In this introductory chapter we want to give a brief survey of results on invariant manifolds and fibers in infinite dimensions as well as describe the aims and scope of this monograph. Recent books by Wiggins [86] and Bronstein and Kopanskii [9] survey invariant manifold results in finite dimensions. Here we will be focusing solely on surveying results in the infinite-dimensional setting.

Charles Li, Stephen Wiggins

2. The Perturbed Nonlinear Schrödinger Equation

Abstract

Consider the perturbatively damped and driven nonlinear Schrödinger equation (PNLS)

$$i{{q}_{t}} = {{q}_{{xx}}} + 2\left[ {|q{{|}^{2}} - {{\omega }^{2}}} \right]q + i\epsilon \left[ { - \alpha q + {{{\hat{D}}}^{2}}q + \Gamma } \right]$$

(2.1.1)

under the even and periodic boundary condition

$$\begin{array}{*{20}{c}} {q( - x) = q(x),} & {q(x + 1) = q(x),} \\ \end{array}$$

where $\omega \in (\pi ,2\pi ),\epsilon \in ( - {{\epsilon }_{0}},{{\epsilon }_{0}})$ is the perturbation parameter, α(> 0) and are real constants. The operator ${{\hat{D}}^{2}}$ is a regularized Laplacian, specifically given by

$${{\hat{D}}^{2}}q \equiv - \sum\limits_{{j = 1}}^{\infty } {{{\beta }_{j}}k_{j}^{2}{{{\hat{q}}}_{j}}\cos {{k}_{j}}x,}$$

where ${{\hat{q}}_{j}}$ is the Fourier transform of q and ${{k}_{j}} \equiv 2\pi j$ The regularizing coefficient β_j is defined by

$${{\beta }_{j}} \equiv \left\{ {\begin{array}{*{20}{c}} \beta \hfill & {for j \leqslant N,} \hfill \\ {{{\alpha }_{*}}k_{j}^{{ - 2}}} \hfill & {for j > N,} \hfill \\ \end{array} } \right.$$

where α_*, and β are positive constants and N is a large fixed positive integer. When, the terms and are perturbatively damping terms; the former is a linear damping, and the latter is a diffusion term. Hence, this regularized Laplacian acts in such a way that it smooths the dissipation at short wavelengths. The reason for this choice is that we will need the flow generated by this infinite dimensional dynamical system to be defined for all time. We will see that the condition ω ∈ (π, 2π) implies that for an appropriate linearization of the unperturbed nonlinear Schrödinger equation (to be discussed shortly), there is precisely one exponentially growing and one exponentially decaying mode (more exponentially growing and decaying modes can be treated without difficulty). ¹ The term is a perturbatively driving term.

Charles Li, Stephen Wiggins

3. Persistent Invariant Manifolds

Abstract

In this section, we establish the existence of certain infinite-dimensional (codimension 1 and 2) invariant manifolds for the enlarged, perturbed system (EPNLS) (2.5.9), using a dichotomy of time scales, i.e., using normal hyperbolicity.

Charles Li, Stephen Wiggins

4. Fibrations of the Persistent Invariant Manifolds

Abstract

We continue to work in ${{\tilde{D}}_{k}}$, where the bumped perturbed flow (2.6.27) is defined. From Proposition 3.1.1, we know the existence of the C ⁿ codimension 1 center-unstable manifold $W_{{{{\delta }_{1}},\delta }}^{{cu}}$, the Cⁿ codimension 1 center-stable manifold $W_{{{{\delta }_{1}},\delta }}^{{cs}}$, and the Cⁿ codimension 2 center manifold ${{W}_{{{{\delta }_{1}},\delta }}}$, under the bumped perturbed flow (2.6.27). More specifically, $W_{{{{\delta }_{1}},\delta }}^{{cu}}$ exists in $\tilde{D}_{k}^{{(1)}}$; moreover, it is overflowing invariant. $W_{{{{\delta }_{1}},\delta }}^{{cs}}$ exists in $\tilde{D}_{k}^{{(2)}}$; moreover, it is inflowing invariant. Then ${{W}_{{{{\delta }_{1}},\delta }}} \equiv W_{{{{\delta }_{1}},\delta }}^{{cu}} \cap W_{{{{\delta }_{1}},\delta }}^{{cs}}$ exists in $\tilde{D}_{k}^{{(2)}}$, and it is inflowing invariant. Since the fibration theorem is concerned with the fiber representations of $W_{{{{\delta }_{1}},\delta }}^{{cu}}$ and $W_{{{{\delta }_{1}},\delta }}^{{cs}}$ with respect to ${{W}_{{{{\delta }_{1}},\delta }}}$ as the base, we have to work in a region where ${{W}_{{{{\delta }_{1}},\delta }}}$ exists. Therefore, we can work only inside $\tilde{D}_{k}^{{(2)}}$. We know that ${{W}_{{{{\delta }_{1}},\delta }}}$ is inflowing invariant in $\tilde{D}_{k}^{{(2)}}$. Next, we will prove the following lemma:

Charles Li, Stephen Wiggins

Backmatter

Title: Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations
Authors: Charles Li
Stephen Wiggins
Publisher: Springer New York
Electronic ISBN: 978-1-4612-1838-8
Print ISBN: 978-1-4612-7307-3
DOI: https://doi.org/10.1007/978-1-4612-1838-8

Springer Professional

Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations

About this book

Table of Contents

Frontmatter

1. Introduction

2. The Perturbed Nonlinear Schrödinger Equation

3. Persistent Invariant Manifolds

4. Fibrations of the Persistent Invariant Manifolds

Backmatter