Symmetries and Integrability of Difference Equations

This volume comprises state-of-the-art articles in discrete integrable systems.

Peter A. Clarkson (Edited by), Frank W. Nijhoff (Edited by)

9780521596992, Cambridge University Press

Paperback, published 4 February 1999

444 pages
22.9 x 15.3 x 2.5 cm, 0.674 kg

'… this book will form an inspiration for further research and so help to establish the links between the various communities working on discrete systems.' European Mathematical Society

There has in recent years been a remarkable growth of interest in the area of discrete integrable systems. Much progress has been made by applying symmetry groups to the study of differential equations, and connections have been made to other topics such as numerical methods, cellular automata and mathematical physics. This volume comprises state of the art articles from almost all the leading workers in this important and rapidly developing area, making it a necessary resource for all researchers interested in discrete integrable systems or related subjects.

Part I. Partial Differential Equations: 1. Discrete linearisable Gambier equations A. K. Common, E. Hesameddini and M. Musette
2. Generalized Bäcklund transformation and new explicit solutions of the two-dimensional Toda equation X.-B. Hu and P. Clarkson
3. Different aspects of relativistic Toda Chain S. Kharchev, A. Mironov, A. Zhedanov
Part II. Integrable Mappings: 4. Integrable symplectic maps A. Fordy
5. An iterative process on quartics and integrable symplectic maps J. P. Francoise, and O. Ragnisco
6. Integrable mappings of KdV type and hyperelliptic addition theorems F. Nijhoff and V. Z. Enolskii
7. R-matrix hierarchies, integrable lattice systems and their integrable discretizations Y. Suris
Part III. Discrete Geometry: 8. Discrete conformal maps and surfaces A. Bobenko
9. The Bäcklund transformation for discrete isothermic surfaces J. Cieslinski
10. Integrable discrete geometry with ruler and compass A. Doliwa
11. Self-dual Einstein spaces and a discrete Tzitzeica equation. A permutability theorem link W. K. Schief
Part IV. Asymptotic Analysis: 12. New solutions of the non-stationary Schrödinger and Kadomtsev-Petviashvili equation M. Ablowitz
13. On asymptotic analysis of the orthogonal polynomials via the Riemann-Hilbert method P. Bleher and A. Its
14. A new spectral transform for solving the continuous and spatially discrete heat equation on simple trees P. C. Bressloff and A. Fokas
Part V. Discrete Painlevé Equations: 15. The discrete Painlevé hierarchy C. Cresswell and N. Joshi
16. Rational solutions to d-PIV J. Hietarinta and K. Kajiwara
17. The discrete Painlevé II equation and the classical special functions K. Kajiwara
18. Freud's equations for orthogonal polynomials as discrete Painlevé equations A. Magnus
Part VI. Symmetries of Difference Equations: 19. An approach to master symmetries of lattice equations B. Fuchssteiner and W. X. Ma
20. Symmetries and generalized symmetries for discrete dynamical systems D. Levi
21. Nonlinear difference equations with superposition formulas P. Winternitz
Part VII. Numerical Methods and Miscellaneous: 22. Generalising the Painlevé truncation: expansions in Riccati pseudopotentials A. Fordy and A. Pickering
23. Symplectic Runge-Kutta schemes W. Oevel
Part VIII. Cellular Automata: 24. Soliton cellular automata C. Gilson
25. Painlevé equations and cellular automata B. Grammaticos and A. Ramani
26. 2+1 Dimensional soliton cellular automaton S. Moriwaki, A. Nagaki, J. Satsuma, T. Tokihiro, M. Torii, D. Takahashi and J. Matsukidaira
Part IX. q-Special Functions and q-Difference Equations: 27. Fourier–Gauss transforms of q-exponential and q-Bessel functions N. Atakishiyev
28. The Wilson bispectral involution: some elementary examples F. A. Grünbaum and L. Haine
29. Factorisation of Macdonald polynomials V. B. Kuztensov and E. K. Skylanin
30. Local Yang-Baxter relations associated with Hirota's discrete equation R. Kashaev
31. Ultra-discrete soliton systems J. Satsuma
32. Schrödinger equation on quantum homogeneous spaces F. Bonechi, R. Giachetti, E. Sorace, M. Tarlini
33. Some algebraic solutions of discrete equations from anticommuting variables C. Viallet
34. Q-combinatorics and quantum integrability A. Volkov.

Subject Areas: Integral calculus & equations [PBKL]