Symmetries and Integrability of Difference Equations

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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.

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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
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 (
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Product Details

Meet the Author

Decio Levi is a researcher in the Faculty of Engineering at the Università 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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