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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- L. Bianchi, Vorlesungen über Differenzialgeometrie, Teubner, Leipzig, 1899.Google Scholar
- A. I. Bobenko and Yu. B. Suris, Discrete Differential Geometry. Integrable Structure, Graduate Studies in Math., vol. 98, Amer. Math. Soc., Providence, RI, 2008.Google Scholar
- I. G. Korepanov, Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics, http://arxiv.org/abs/solv-int/9506003.
- V. Papageorgiou, A. Tongas, and A. Veselov, “Yang-Baxter maps and symmetries of integrable equations on quad-graphs,” J. Math. Phys, 47:8 (2006), 083502.Google Scholar
- A. P. Veselov, “Yang-Baxter maps: dynamical point of view,” in: Combinatorial Aspect of Integrable Systems, MSJ Mem., vol. 17, Math. Soc. Japan, Tokyo, 2007, 145–167.Google Scholar
- C. Viallet, Algebraic Entropy for Lattice Equations, http://arxiv.org/abs/math-ph/0609043.
- E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 1927, reprinted in 1996.Google Scholar