Discrete nonlinear hyperbolic equations. Classification of integrable cases

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• 作者：V. E. Adler ; A. I. Bobenko and Yu. B. Suris
• 关键词：integrability ; quad ; graph ; multidimensional consistency ; zero curvature representation ; B?cklund transformation ; Bianchi permutability ; M?bius transformation
• 刊名：Functional Analysis and Its Applications
• 出版年：2009
• 出版时间：March, 2009
• 年：2009
• 卷：43
• 期：1
• 页码：3-17
• 全文大小：220.2 KB

文摘

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.