Discrete nonlinear hyperbolic equations. Classification of integrable cases

V. E. Adler, A. I. Bobenko, Yu B. Suris

Research output: Contribution to journalArticlepeer-review

108 Scopus citations


We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on 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 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.

Original languageEnglish
Pages (from-to)3-17
Number of pages15
JournalFunctional Analysis and its Applications
Issue number1
StatePublished - Mar 2009


  • Bianchi permutability
  • Bäcklund transformation
  • Integrability
  • Multidimensional consistency
  • Möbius transformation
  • Quad-graph
  • Zero curvature representation


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