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 ℤN. 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.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 43, No. 1, pp. 3–21, 2009
Original Russian Text Copyright © by V. E. Adler, A. I. Bobenko, and Yu. B. Suris
The first author supported by the DFG Research Unit 565 “Polyhedral Surfaces” and by RFBR grant 04-01-00403. The second author supported in part by the DFG Research Unit 565 “Polyhedral Surfaces.” The third author supported in part by the ESF Scientific Programme “Methods of Integrable Systems, Geometry, Applied Mathematics” (MISGAM).
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Adler, V.E., Bobenko, A.I. & Suris, Y.B. Discrete nonlinear hyperbolic equations. Classification of integrable cases. Funct Anal Its Appl 43, 3–17 (2009). https://doi.org/10.1007/s10688-009-0002-5
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DOI: https://doi.org/10.1007/s10688-009-0002-5