Nonlinear integrable lattices with three independent variables

Abstract

We suggest an algorithm for deriving nonlinear integrable equations of the form
$$u^{j+1}_{n,x}=F(u^{j}_{n,x},u^{j+1}_{n},u^{j}_{n+1},u^{j}_{n},u^{j+1}_{n-1})$$
with three independent variables; the algorithm uses the known list of Toda type integrable equations. The algorithm is based on the Darboux integrable finite field reductions, construction of a complete set of characteristic integrals and dicretization via integrals.