Classification of integrable equations on quad-graphs. The consistency approach

Details Classification of integrable equations on quad-graphs. The consistency approach Classification of integrable equations on quad-graphs. The consistency approach

2002-02-11

arxiv.org/abs/nlin/0202024v2

Commun. Math. Phys., 2003, vol. 233, 513-543

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Scientific paper

10.1007/s00220-002-0762-8

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 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.

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