Integrable systems on quad-graphs

TL;DR

This paper explores integrable systems on quad-graphs, establishing a connection with Toda-type systems, and introduces an algorithmic method for deriving zero curvature representations based on three-dimensional consistency, with geometric insights via circle patterns.

Contribution

It introduces a systematic procedure for deriving zero curvature representations for integrable systems on quad-graphs, linking them to Toda systems and geometric circle patterns.

Findings

Established relation between integrable systems on quad-graphs and Toda systems.

Developed an algorithmic method for zero curvature representations.

Provided geometric interpretation through circle patterns.

Abstract

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.