Miura-Type Transformations for Integrable Lattices in 3D

This article studies a class of integrable semi-discrete equations with one continuous and two discrete independent variables. At present, in the literature there are nine integrable equations of the form un+1,xj=f(un,xj,unj+1,unj,un+1j,un+1j-1) up to point transformations. An efficient method based on some relation that generalizes the notion of the local conservation law is proposed for searching for Miura-type transformations relating to semi-discrete equations in 3D. The efficiency of the method is illustrated with the equations from the list. For one of the equations, which is little studied, the continuum limit is calculated. For this equation, the problem of finite-field reductions in the form of Darboux integrable systems of equations of a hyperbolic type is discussed. For reductions of small orders, N=1 and N=2, complete sets of characteristic integrals are presented. Note that the existence of characteristic integrals makes it possible to construct particular solutions to the original lattice. For the case N=1, an explicit solution was found in this paper. A new semi-discrete equation is found that lies beyond the considered class. For this equation, the Lax pair is presented.