Overview
- Systematic Hamiltonian recipe of integrable discretization
- Unique large collection of results about discrete integrable systems
- Unfied treatment of the correspondent continuous integrable systems
Part of the book series: Progress in Mathematics (PM, volume 219)
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Table of contents (27 chapters)
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Front Matter
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General Theory
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Front Matter
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Lattice Systems
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Front Matter
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Keywords
About this book
The book explores the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property? This question (related in spirit to such a modern branch of numerical analysis as geometric integration) is treated in the book as an immanent part of the theory of integrable systems, also commonly termed as the theory of solitons.
Among several possible approaches to this theory, the Hamiltonian one is chosen as the guiding principle. A self-contained exposition of the Hamiltonian (r-matrix, or "Leningrad") approach to integrable systems is given, culminating in the formulation of a general recipe for integrable discretization of r-matrix hierarchies. After that, a detailed systematic study is carried out for the majority of known discrete integrable systems which can be considered as discretizations of integrable ordinary differential or differential-difference (lattice) equations. This study includes, in all cases, a unified treatment of the correspondent continuous integrable systems as well. The list of systems treated in the book includes, among others: Toda and Volterra lattices along with their numerous generalizations (relativistic, multi-field, Lie-algebraic, etc.), Ablowitz-Ladik hierarchy, peakons of the Camassa-Holm equation, Garnier and Neumann systems with their various relatives, many-body systems of the Calogero-Moser and Ruijsenaars-Schneider type, various integrable cases of the rigid body dynamics. Most of the results are only available from recent journal publications, many of them are new.
Thus, the book is a kind of encyclopedia on discrete integrable systems. It unifies the features of a research monograph and a handbook. It is supplied with an extensive bibliography and detailed bibliographic remarks at the end of each chapter. Largely self-contained, it will beaccessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems. Also those involved in real numerical calculations or modelling with integrable systems will find it very helpful.
Authors and Affiliations
Bibliographic Information
Book Title: The Problem of Integrable Discretization
Book Subtitle: Hamiltonian Approach
Authors: Yuri B. Suris
Series Title: Progress in Mathematics
DOI: https://doi.org/10.1007/978-3-0348-8016-9
Publisher: Birkhäuser Basel
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eBook Packages: Springer Book Archive
Copyright Information: Springer Basel AG 2003
Hardcover ISBN: 978-3-7643-6995-8Published: 20 June 2003
Softcover ISBN: 978-3-0348-9404-3Published: 05 November 2012
eBook ISBN: 978-3-0348-8016-9Published: 06 December 2012
Series ISSN: 0743-1643
Series E-ISSN: 2296-505X
Edition Number: 1
Number of Pages: MLXX, 21
Topics: Order, Lattices, Ordered Algebraic Structures, Dynamical Systems and Ergodic Theory, Computational Mathematics and Numerical Analysis, Theoretical, Mathematical and Computational Physics, Numerical and Computational Physics, Simulation, Solid State Physics