Skip to main content

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


 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.

This is a preview of subscription content, access via your institution.

Author information

Authors and Affiliations


Additional information

Received: 14 February 2002 / Accepted: 22 September 2002 Published online: 8 January 2003

Acknowledgements. This research was partly supported by DFG (Deutsche Forschungsgemeinschaft) in the frame of SFB 288 ``Differential Geometry and Quantum Physics''. V.A. was also supported by the RFBR grant 02-01-00144. He thanks TU Berlin for warm hospitality during the visit when part of this work has been fulfilled.

Communicated by L. Takhtajan

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Adler, V., Bobenko, A. & Suris, Y. Classification of Integrable Equations on Quad-Graphs. The Consistency Approach. Commun. Math. Phys. 233, 513–543 (2003).

Download citation

  • Issue Date:

  • DOI:


  • Integrable System
  • Spectral Parameter
  • Discrete System
  • Symplectic Structure
  • Exhaustive List