Semiclassical soliton ensembles for the focusing nonlinear Schrodinger equation / Spyridon Kamvissis, Kenneth D.T-R McLaughlin, Peter D. Miller.

Format:

Book

Author/Creator:

Kamvissis, Spyridon.

Contributor:

McLaughlin, K. T-R (Kenneth T-R), 1969-

Miller, Peter D. (Peter David), 1967-

Series:

Annals of mathematics studies ; no. 154.

Annals of mathematics studies ; no. 154

Language:

English

Subjects (All):

Schrödinger equation.

Wave mechanics.

Physical Description:

1 online resource (280 p.)

Edition:

Course Book

Place of Publication:

Princeton, NJ : Princeton University Press, c2003.

Language Note:

English

Summary:

This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

Contents:

Frontmatter

Contents

Figures and Tables

Preface

Chapter 1. Introduction and Overview

Chapter 2. Holomorphic Riemann-Hilbert Problems for Solitons

Chapter 3. Semiclassical Soliton Ensembles

Chapter 4. Asymptotic Analysis of the Inverse Problem

Chapter 5. Direct Construction of the Complex Phase

Chapter 6. The Genus - Zero Ansatz

Chapter 7. The Transition to Genus Two

Chapter 8. Variational Theory of the Complex Phase

Chapter 9. Conclusion and Outlook

Appendix A. H¨older Theory of Local Riemann-Hilbert Problems

Appendix B. Near-Identity Riemann-Hilbert Problems in L2

Bibliography

Index

Notes:

Description based upon print version of record.

Includes bibliographical references (p. [255]-258) and index.

ISBN:

1-299-44345-1

1-4008-3718-9

OCLC:

845252685