Discrete nonlinear hyperbolic equations. Classification of integrable cases

2007/05/11 by Vsevolod E. Adler, Alexander I. Bobenko, Yuri B. Suris · 3 citations
Physics and Astronomy · #nlin.SI

paper · pdf

published as Funkt. Analiz Prilozh., 2009, 43, p. 3-21; English translation: Funct. Anal. Appl., 2009, 43, p. 3-17. · 19 pages

arxiv created 2007/05/11 · arxiv updated 2009/12/01

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(x₁,x₂,x₃,x₄)=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.

Cited by

• Special solutions to five autonomous integrable partial difference equations via the third and sixth Painlevé equations and the Garnier system in two variables
• On consistency around a 3 × 3× 3 cube and Q3 analogue of the lattice Boussinesq equation
• On the symmetry classification of integrable chains in 3D. Darboux-integrable reductions and their higher symmetries