Discrete systems and integrability
著者
書誌事項
Discrete systems and integrability
（Cambridge texts in applied mathematics）
Cambridge University Press, 2016
- : hardback
- : pbk
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注記
Includes bibliographical references (p. 411-439) and index
内容説明・目次
内容説明
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 Backlund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Pade approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painleve 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.
目次
- 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 Painleve equations
- 12. Lagrangian multiform theory
- Appendix A. Elementary difference calculus and difference equations
- Appendix B. Theta functions and elliptic functions
- Appendix C. The continuous Painleve equations and the Garnier system
- Appendix D. Some determinantal identities
- References
- Index.
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