© 2003

The Problem of Integrable Discretization: Hamiltonian Approach


Part of the Progress in Mathematics book series (PM, volume 219)

Table of contents

  1. Front Matter
    Pages i-xxi
  2. General Theory

    1. Front Matter
      Pages 1-1
    2. Yuri B. Suris
      Pages 3-50
    3. Yuri B. Suris
      Pages 51-100
  3. Lattice Systems

    1. Front Matter
      Pages 101-101
    2. Yuri B. Suris
      Pages 103-172
    3. Yuri B. Suris
      Pages 173-229
    4. Yuri B Suris
      Pages 231-262
    5. Yuri B. Suris
      Pages 263-320
    6. Yuri B. Suris
      Pages 321-353
    7. Yuri B. Suris
      Pages 445-454
    8. Yuri B. Suris
      Pages 455-481
    9. Yuri B. Suris
      Pages 507-532
    10. Yuri B. Suris
      Pages 533-565
    11. Yuri B. Suris
      Pages 605-641

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 be accessible 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.


Lattice Matrix computational and mathematical physics dynamical systems numerical analysis

Authors and affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Bibliographic information