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Communications in Mathematical Physics

, Volume 233, Issue 3, pp 513–543 | Cite as

Classification of Integrable Equations on Quad-Graphs. The Consistency Approach

  • V.E. Adler
  • A.I. Bobenko
  • Yu.B. Suris

Abstract:

 A classification of discrete integrable systems on quad–graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three–dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature representation with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three–leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three–dimensional integrable systems and to the quantum context are also discussed.

Keywords

Integrable System Spectral Parameter Discrete System Symplectic Structure Exhaustive List 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • V.E. Adler
    • 1
  • A.I. Bobenko
    • 2
  • Yu.B. Suris
    • 2
  1. 1.Landau Institute of Theoretical Physics, 12 Institutsky pr., 142432 Chernogolovka, Russia. E-mail: adler@itp.ac.ruRU
  2. 2.Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany. E-mail: bobenko@math.tu-berlin.de; suris@sfb288.math.tu-berlin.deDE

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