Discrete and Continuous Nonlinear Schrödinger Systems

✍ Scribed by M. J. Ablowitz, B. Prinari, A. D. Trubatch

• Publisher: Cambridge University Press
• Year: 2004
• Tongue: English
• Leaves: 268
• Series: London Mathematical Society Lecture Note Series
• Category: Library

✦ Synopsis

Over the past thirty years significant progress has been made in the investigation of nonlinear waves--including "soliton equations", a class of nonlinear wave equations that arise frequently in such areas as nonlinear optics, fluid dynamics, and statistical physics. The broad interest in this field can be traced to understanding "solitons" and the associated development of a method of solution termed the inverse scattering transform (IST). The IST technique applies to continuous and discrete nonlinear Schrödinger (NLS) equations of scalar and vector type. This work presents a detailed mathematical study of the scattering theory, offers soliton solutions, and analyzes both scalar and vector soliton interactions. The authors provide advanced students and researchers with a thorough and self-contained presentation of the IST as applied to nonlinear Schrödinger systems.

✦ Table of Contents

Contents......Page 6
Preface......Page 8
1.1 Solitons and soliton equations......Page 12
1.2 The inverse scattering transform– Overview......Page 14
1.3 Nonlinear Schr¨odinger systems......Page 16
1.4 Physical applications......Page 20
1.5 Outline of the work......Page 27
2.1 Overview......Page 29
2.2 The inverse scattering transformfor NLS......Page 30
2.3 Soliton solutions......Page 50
2.4 Conserved quantities and Hamiltonian structure......Page 53
3.1 Overview......Page 57
3.2 The inverse scattering transformfor IDNLS......Page 59
3.3 Soliton solutions......Page 94
3.4 Conserved quantities and Hamiltonian structure......Page 98
4.1 Overview......Page 101
4.2 The inverse scattering transformfor MNLS......Page 102
4.3 Soliton solutions......Page 124
4.4 Conserved quantities and Hamiltonian structure......Page 139
5.1 Overview......Page 141
5.2 The inverse scattering transformfor IDMNLS......Page 144
5.3 Soliton solutions......Page 196
5.4 Conserved quantities......Page 212
Appendix A: Summation by parts formula......Page 215
Appendix B: Transmission of the Jost function through a localized potential......Page 217
C.1 Introduction......Page 219
C.2 Direct scattering problem......Page 221
C.3 Inverse scattering problem......Page 231
C.4 Time evolution and solitons for the Toda lattice and nonlinear ladder network......Page 237
D.1 Continuous NLS systems with a potential term......Page 240
D.2 Discrete NLS systems with a potential term......Page 244
Appendix E: NLS systems in the limit of large amplitudes......Page 250
Bibliography......Page 254
Index......Page 266