The Problem of Integrable Discretization: Hamiltonian Approach

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Birkhäuser, Dec 6, 2012 - Mathematics - 21 pages

An exploration of 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. 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 be accessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems.

 

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Contents

Hamiltonian Mechanics
3
Explicit Discretizations of Newtonian Toda Systems
10
Rmatrix Hierarchies
51
Lattice Systems
98
Volterra Lattice
173
Newtonian Equations of the Toda Type
231
Relativistic Toda Lattice
263
Relativistic Volterra Lattice
321
Multifield Relativistic Volterra Systems
567
Bogoyavlensky Lattices
605
AblowitzLadik Hierarchy
643
Peakons System
689
Standardlike Discretizations
701
Liealgebraic Toda Systems
711
Garnier System
777
HénonHeiles System
817

Newtonian Equations of the Relativistic Toda Type
355
Explicit Discretizations for Toda Systems
409
BruschiRagnisco Lattice
445
Multifield Todalike Systems
455
Multifield Relativistic Toda Systems
483
BelovChaltikian Lattices
507
Multifield Volterralike Systems
533
Neumann System
823
Liealgebraic Generalizations of the Garnier Systems
877
Integrable Cases of Rigid Body Dynamics
897
Systems of CalogeroMoser Type
947
Bibliography
1001
List of Notations
1043
Copyright

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