Discrete nonlinear hyperbolic equations. Classification of integrable cases
Exactly Solvable and Integrable Systems
2009-06-12 v1
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.
Cite
@article{arxiv.0705.1663,
title = {Discrete nonlinear hyperbolic equations. Classification of integrable cases},
author = {Vsevolod E. Adler and Alexander I. Bobenko and Yuri B. Suris},
journal= {arXiv preprint arXiv:0705.1663},
year = {2009}
}