## Abstract

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 language | English |
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Pages (from-to) | 3-17 |

Number of pages | 15 |

Journal | Functional Analysis and its Applications |

Volume | 43 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2009 |

## Keywords

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