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

By V. E. Adler, Alexander I. Bobenko, Yuri B. Suris. 2002

168 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2db517580f56dc2a1c1cbf2dc955f6f620337947

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In Geometry and Topology

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Citations in the corpus

Burchnall-Chaundy Theory for q-Difference Operators and q-Deformed Heisenberg Algebras (2003, background): “In fact the 3D consistency condition is the exact analog of the Yang-Baxter relation for the problems with fields on vertices (see [5]).”
Symmetries of Integrable Difference Equations on the Quad‐Graph (2007, unknown): “Adler, Bobenko & Suris classified integrable equations on the quad-graph by using the observation that Lax pairs arise from consistency on the cube [1].”