Discrete Systems and Integrability

A first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.

J. Hietarinta (Author), N. Joshi (Author), F. W. Nijhoff (Author)

9781107042728, Cambridge University Press

Hardback, published 1 September 2016

458 pages, 68 b/w illus. 2 tables 98 exercises
25.3 x 18 x 3 cm, 0.96 kg

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.

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.

Subject Areas: Differential calculus & equations [PBKJ], Calculus & mathematical analysis [PBK]