Symmetries and Integrability of Difference Equations

A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike.

Decio Levi (Edited by), Peter Olver (Edited by), Zora Thomova (Edited by), Pavel Winternitz (Edited by)

9780521136587, Cambridge University Press

Paperback, published 23 June 2011

360 pages, 40 b/w illus.
22.8 x 15.1 x 1.8 cm, 0.51 kg

'The book can certainly be recommended to everyone interested in having a comprehensive overview of the current state of the art in this fascinating and important research field.' Frans Cantrijn, Mathematical Reviews

Difference equations are playing an increasingly important role in the natural sciences. Indeed many phenomena are inherently discrete and are naturally described by difference equations. Phenomena described by differential equations are therefore approximations of more basic discrete ones. Moreover, in their study it is very often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference equations. This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference ones. Each of the eleven chapters is a self-contained treatment of a topic, containing introductory material as well as the latest research results. The book will be welcomed by graduate students and researchers seeking an introduction to the field. As a survey of the current state of the art it will also serve as a valuable reference.

1. Lagrangian and Hamiltonian formalism for discrete equations: symmetries and first integrals V. Dorodnitsyn and R. Kozlov
2. Painlevé equations: continuous, discrete and ultradiscrete B. Grammaticos and A. Ramani
3. Definitions and predictions of integrability for difference equations J. Hietarinta
4. Orthogonal polynomials, their recursions, and functional equations M. E. H. Ismail
5. Discrete Painlevé equations and orthogonal polynomials A. Its
6. Generalized Lie symmetries for difference equations D. Levi and R. I. Yamilov
7. Four lectures on discrete systems S. P. Novikov
8. Lectures on moving frames P. J. Olver
9. Lattices of compact semisimple Lie groups J. Patera
10. Lectures on discrete differential geometry Yu. B Suris
11. Symmetry preserving discretization of differential equations and Lie point symmetries of differential-difference equations P. Winternitz.

Subject Areas: Mathematical physics [PHU], Differential calculus & equations [PBKJ]