Description
Book SynopsisThis 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 BÃcklund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Padà approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painlevà 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 exer
Table of ContentsPreface; 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.
