Symmetries and Integrability of Difference Equations

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.

Table of Contents:
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.