The Problem of Integrable Discretization : Hamiltonian Approach
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 t...
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Format:  eBook 
Language:  English 
Published: 
Basel
Birkhäuser Basel
2003, 2003

Edition:  1st ed. 2003 
Series:  Progress in Mathematics

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Online Access:  
Collection:  Springer Book Archives 2004  Collection details see MPG.ReNa 
Summary:  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 selfcontained exposition of the Hamiltonian (rmatrix, or "Leningrad") approach to integrable systems is given, culminating in the formulation of a general recipe for integrable discretization of rmatrix hierarchies. 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 selfcontained, it will be accessible to graduate and postgraduate 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 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 differentialdifference (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, multifield, Liealgebraic, etc.), AblowitzLadik hierarchy, peakons of the CamassaHolm equation, Garnier and Neumann systems with their various relatives, manybody systems of the CalogeroMoser and RuijsenaarsSchneider 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. 

Physical Description:  XXI, 1070 p online resource 
ISBN:  9783034880169 