Discrete nonlinear hyperbolic equations. Classification of integrable cases

Abstract

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of equations is understood as 3D-consistency. The latter is a possibility to consistently impose equations of the same type on all the faces of a three-dimensional cube. This allows to set these equations also on multidimensional lattices Z^N. We classify integrable equations with complex fields x, and Q affine-linear with respect to all arguments. The method is based on analysis of singular solutions.

19 pages

References in corpus (3)

• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Yang-Baxter maps: dynamical point of view
• Algebraic Entropy for lattice equations

Cited by in corpus (1)

• On a nonlinear integrable difference equation on the square 3D-inconsistent