Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2001-10-04
Internat. Math. Research Notices, 2002, Nr. 11, p.573-611
Nonlinear Sciences
Exactly Solvable and Integrable Systems
29 pages, 11 figures
Scientific paper
We consider general integrable systems on graphs as discrete flat connections with the values in loop groups. We argue that a certain class of graphs is of a special importance in this respect, namely quad-graphs, the cellular decompositions of oriented surfaces with all two-cells being quadrilateral. We establish a relation between integrable systems on quad-graphs and discrete systems of the Toda type on graphs. We propose a simple and general procedure for deriving discrete zero curvature representations for integrable systems on quad-graphs, based on the principle of the three-dimensional consistency. Thus, finding a zero curvature representation is put on an algorithmic basis and does not rely on the guesswork anymore. Several examples of integrable systems on quad-graphs are considered in detail, their geometric interpretation is given in terms of circle patterns.
Bobenko Alexander I.
Suris Yuri B.
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