Functional Analysis and Its Applications

, Volume 43, Issue 1, pp 3–17

# Discrete nonlinear hyperbolic equations. Classification of integrable cases

• A. I. Bobenko
• Yu. B. Suris
Article

## Abstract

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ℤ2. The fields are associated with the vertices and an equation of the form Q(x 1, x 2, x 3, x 4) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices ℤ N . We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.

## Key words

integrability quad-graph multidimensional consistency zero curvature representation Bäcklund transformation Bianchi permutability Möbius transformation

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