A partial difference equation (PΔE) is a fully discretized version of a partial differential equation. The talk focuses on 2-dimensional nonlinear PΔEs which are completely integrable, i.e., they admit a Lax representation.
Based on work by Nijhoff, Bobenko and Suris, a method to compute Lax pairs will be presented. The method is largely algorithmic and can be implemented in the syntax of computer algebra systems, such as Mathematica and Maple.
A Mathematica program will be presented that automatically computes Lax pairs for a variety of 2-dimensional PΔEs, including lattice versions of the potential Korteweg-de Vries (KdV) equations, the modified KdV and sine-Gordon equations, as well as lattices derived by Adler, Bobenko, and Suris.