Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases

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.