Discrete nonlinear hyperbolic equations. Classification of integrable cases

TL;DR

This paper classifies integrable discrete nonlinear hyperbolic equations on quad-graphs, using 3D-consistency and singular solutions analysis, expanding understanding of their structure and integrability conditions.

Contribution

It provides a complete classification of affine-linear, complex-field hyperbolic equations on quad-graphs based on 3D-consistency criteria.

Findings

Identified all integrable cases of affine-linear equations on quad-graphs.

Established 3D-consistency as a key integrability criterion.

Developed a method based on singular solutions analysis.

Abstract

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of equations is understood as 3D-consistency. The latter is a possibility to consistently impose equations of the same type on all the faces of a three-dimensional cube. This allows to set these equations also on multidimensional lattices Z^N. We classify integrable equations with complex fields x, and Q affine-linear with respect to all arguments. The method is based on analysis of singular solutions.