Symmetries and Integrability of Difference Equations

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Overview

This book is devoted to a topic that has undergone rapid and fruitful development over the last few years: symmetries and integrability of difference equations and $q$-difference equations and the theory of special functions that occur as solutions of such equations. Techniques that have been traditionally applied to solve linear and nonlinear differential equations are now being successfully adapted and applied to discrete equations. This volume is based on contributions made by leading experts in the field during the workshop on Symmetries and Integrability of Difference Equations held in Esterel, Quebec, in May 1994. Giving an up-to-date review of the current status of the field, the book treats these specific topics: Lie group and quantum group symmetries of difference and $q$-difference equations, integrable and nonintegrable discretizations of continuous integrable systems, integrability of difference equations, discrete Painleve property and singularity confinement, integrable mappings, applications in statistical mechanics and field theories, Yang-Baxter equations, $q$-special functions and discrete polynomials, and $q$-difference integrable systems.

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Editorial Reviews

From the Publisher
"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
Booknews
Results not yet published elsewhere and overviews of various aspects are among the 42 papers by researchers from different disciplines who use discrete integrable systems, that is systems that can be described by ordinary or partial difference equations and can be solved using exact methods. The themes include Lie-type symmetries of differential- difference and difference-difference equations, integrable differential-difference equations, fully discrete integrable equations from multi-dimensional lattices to cellular automata, discrete geometry and integrability, and continuous and discrete Painlev<'e> equations and related problems. There is no index. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

Meet the Author

Decio Levi is a researcher in the Faculty of Engineering at the Universit... degli Studi Roma Tre.

Peter Olver is a Professor and currently Head of the School of Mathematics at the University of Minnesota.

Zora Thomova is an Associate Professor of Mathematics at the State University of New York - Institute of Technology.

Pavel Winternitz is a Professor in the Department of Mathematics and Statistics at the Université de Montréal.

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Table of Contents

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.

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