Discrete and Continuous Nonlinear Schrödinger Systems (London Mathematical Society Lecture Note, Vol. 302) (London Mathematical Society Lecture Note Series, Series Number 302)

In recent years there have been important and far reaching developments in the study of nonlinear waves and a class of nonlinear wave equations which arise frequently in applications. The wide interest in this field comes from the understanding of special waves called 'solitons' and the associated development of a method of solution to a class of nonlinear wave equations termed the inverse scattering transform (IST). Before these developments, very little was known about the solutions to such 'soliton equations'. The IST technique applies to both continuous and discrete nonlinear Schrödinger equations of scalar and vector type. Also included is the IST for the Toda lattice and nonlinear ladder network, which are well-known discrete systems. This book, first published in 2003, presents the detailed mathematical analysis of the scattering theory; soliton solutions are obtained and soliton interactions, both scalar and vector, are analyzed. Much of the material is not available in the previously-published literature. 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 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 Schrḏinger (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 Schrḏinger Systems. 1. Introduction -- 2. Nonlinear Schrḏinger Equation (nls) -- 3. Integrable Discrete Nonlinear Schrḏinger Equation (idnls) -- 4. Matrix Nonlinear Schrḏinger Equation (mnls) -- 5 Integrable Discrete Matrix Nls Equation (idmnls). M.j. Ablowitz, B. Prinari, A.d. Trubatch. Includes Bibliographical References (p. 243-254) And Index. "In recent years there have been important and far reaching developments in the study of nonlinear waves and a class of nonlinear wave equations which arise frequently in applications. The wide interest in this field comes from the understanding of special waves called 'solitons' and the associated development of a method of solution to a class of nonlinear wave equations termed the inverse scattering transform (IST). Before these developments, very little was known about the solutions to such 'soliton equations'. The IST technique applies to both continuous and discrete nonlinear Schrodinger equations of scalar and vector type. Also included is the IST for the Toda lattice and nonlinear ladder network, which are well-known discrete systems. This book presents the detailed mathematical analysis of the scattering theory; soliton solutions are obtained and soliton interactions, both scalar and vector, are analyzed. Much of the material is not available in the previously-published literature."--BOOK JACKET 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. In this book, the detailed mathematical analysis of the scattering theory is presented, soliton solutions are obtained and soliton interactions, both scalar and vector, are analyzed. Many details presented, including aspects of the scattering theory, solution methodology, soliton solutions and vector soliton interactions are not available in the previously-published literature Ever since the observation of the "great wave of translation" in water waves, by J. Scott Russell in 1834 [146, 147] while he rode on horseback near a narrow canal in Edinburgh, localized (nonoscillatory) solitary waves have been known to researchers studying wave dynamics.