Geodesic
Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 1, pp. 3-21.

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We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on $\mathbb{Z}^2$. 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 $\mathbb{Z}^N$. 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.
Keywords: integrability, quad-graph, multidimensional consistency, zero curvature representation, Bäcklund transformation, Bianchi permutability
Mots-clés : Möbius transformation. 
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V. E. Adler; A. I. Bobenko; Yu. B. Suris. Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 1, pp. 3-21. http://geodesic.mathdoc.fr/item/FAA_2009_43_1_a0/

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