Discrete nonlinear hyperbolic equations. Classification of integrable cases
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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 wordsintegrability quad-graph multidimensional consistency zero curvature representation Bäcklund transformation Bianchi permutability Möbius transformation
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