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This article is cited in 81 scientific papers (total in 81 papers)
Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases
V. E. Adlera, A. I. Bobenkob, Yu. B. Surisc 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
Bibliographic databases:
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
Citation in format AMSBIB
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