Discrete Systems and Integrability
Discrete Systems and Integrability
N. Joshi, University of Sydney
F. W. Nijhoff, University of Leeds
This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and Bäcklund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Padé approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painlevé equations; and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the numerous exercises and the thorough list of references will benefit upper-level undergraduate, and beginning graduate students as well as researchers from other disciplines.
- A generous and up-to-date bibliography allows the reader to use the text as a source for branching out into more specialized directions, according to their own interests
- Readers will more easily absorb the ideas and techniques by working through explicit computations built up on the basis of key examples while at the same time being presented with the general theory
- Based on tried and tested lecture notes - the material has been used for teaching undergraduate courses and is tailored for student use
Product details
September 2016Paperback
9781107669482
458 pages
246 × 173 × 25 mm
0.82kg
68 b/w illus. 2 tables 98 exercises
Available
Table of Contents
- Preface
- 1. Introduction to difference equations
- 2. Discrete equations from transformations of continuous equations
- 3. Integrability of P∆Es
- 4. Interlude: lattice equations and numerical algorithms
- 5. Continuum limits of lattice P∆Es
- 6. One-dimensional lattices and maps
- 7. Identifying integrable difference equations
- 8. Hirota's bilinear method
- 9. Multi-soliton solutions and the Cauchy matrix scheme
- 10. Similarity reductions of integrable P∆Es
- 11. Discrete Painlevé equations
- 12. Lagrangian multiform theory
- Appendix A. Elementary difference calculus and difference equations
- Appendix B. Theta functions and elliptic functions
- Appendix C. The continuous Painlevé equations and the Garnier system
- Appendix D. Some determinantal identities
- References
- Index.
