Soliton Equations and Their Algebro-Geometric Solutions: Volume 2, (1+1)-Dimensional Discrete Models (Cambridge Studies in Advanced Mathematics, Series Number 114)

As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results. Cover......Page 1 Half-title......Page 3 Series-title......Page 4 Title......Page 5 Copyright......Page 6 Contents......Page 9 Acknowledgments......Page 11 Introduction......Page 13 1.1 Contents......Page 37 1.2 The Toda Hierarchy, Recursion Relations, Lax Pairs, and Hyperelliptic Curves......Page 38 1.3 The Stationary Toda Formalism......Page 53 1.4 The Stationary Toda Algebro-Geometric Initial Value Problem......Page 84 1.5 The Time-Dependent Toda Formalism......Page 96 1.6 The Time-Dependent Toda Algebro-Geometric Initial Value Problem......Page 115 1.7 Toda Conservation Laws and the Hamiltonian Formalism......Page 129 1.8 Notes......Page 157 2.1 Contents......Page 173 2.2 The KM Hierarchy and its Relation to the Toda Hierarchy......Page 174 2.3 The Stationary KM Formalism......Page 184 2.4 The Time-Dependent KM Formalism......Page 190 2.5 Notes......Page 193 3.1 Contents......Page 198 3.2 The Ablowitz–Ladik Hierarchy, Recursion Relations, Zero-Curvature Pairs, and Hyperelliptic Curves......Page 199 3.3 Lax Pairs for the Ablowitz–Ladik Hierarchy......Page 214 3.4 The Stationary Ablowitz–Ladik Formalism......Page 232 3.5 The Stationary Ablowitz–Ladik Algebro-Geometric Initial Value Problem......Page 248 3.6 The Time-Dependent Ablowitz–Ladik Formalism......Page 261 3.7 The Time-Dependent Ablowitz–Ladik Algebro-Geometric Initial Value Problem......Page 279 3.8 Ablowitz–Ladik Conservation Laws and the Hamiltonian Formalism......Page 293 3.9 Notes......Page 326 Appendix A: Algebraic Curves and Their Theta Functions in a Nutshell......Page 336 Appendix B: Hyperelliptic Curves of the Toda-Type......Page 365 Appendix C: Asymptotic Spectral Parameter Expansions and Nonlinear Recursion Relations......Page 377 Appendix D: Lagrange Interpolation......Page 397 List of Symbols......Page 407 Bibliography......Page 410 Index......Page 435 Errata and Addenda for Volume......Page 438 The focus of this book is on algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions, also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces). The presentation is rigorous, detailed, and self-contained, with ample background material provided in various appendices. Detailed notes for each chapter together with an exhaustive bibliography enhance the presentation offered in the main text As a partner to Volume 1: Dimensional Continuous Models, this book provides a self-contained introduction to solition equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. V. 1. (1 + 1)-dimensional Continuous Models -- Volume 2. (1 + 1)-dimensional Discrete Models. Fritz Gesztesy, Helge Holden. Vol. 2 Written By Fritz Gesztesy And Three Others. Includes Bibliographical References (volume 1, Pages 469-499) And Index. Mode Of Access: World Wide Web.