V. E. Adler, A. I. Bobenko, Yu. B. Suris, “Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases”, Funktsional. Anal. i Prilozhen., 43:1 (2009), 3–21; Funct. Anal. Appl., 43:1 (2009), 3

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Funktsional. Anal. i Prilozhen., 2009, Volume 43, Issue 1, Pages 3–21 (Mi faa2936)

This article is cited in 79 scientific papers (total in 79 papers)

Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases

V. E. Adler^a, A. I. Bobenko^b, Yu. B. Suris^c

^a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
^b Institut für Mathematik, Technische Universität Berlin
^c Zentrum Mathematik, Technische Universität München

Abstract: 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, Möbius transformation

DOI: https://doi.org/10.4213/faa2936

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English version:
Functional Analysis and Its Applications, 2009, 43:1, 3–17

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UDC: 517.962.24+517.965+517.957+517.958
Received: 04.06.2007

Citation: V. E. Adler, A. I. Bobenko, Yu. B. Suris, “Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases”, Funktsional. Anal. i Prilozhen., 43:1 (2009), 3–21; Funct. Anal. Appl., 43:1 (2009), 3–17

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