Belov–Chaltikian and Blaszak–Marciniak lattice equations: Recursion operators and factorization

Abstract

A systematic investigation on the construction of recursion operators for partial differential–difference equations (PDDEs) with two independent variables (one continuous and one discrete) using its generalized symmetries is presented. Also it is explained how to factorize the obtained recursion operators. The applicability of the above procedure have been illustrated for the relativistic toda (RT), Belov–Chaltikian (BC) and Blaszak–Marciniak (BM) lattice equations and shown that the former two lattice equations admit (2×2) matrix recursion operators while the latter one possesses a (3×3) matrix recursion operator. Furthermore, the constructed recursion operators can be written as a factor of 2 distinct invertible matrix operators in each of the lattice equations. It is also proved explicitly that the factorized operators are Hamiltonian and hence RT, BC and BM lattice equations are bi-Hamiltonian systems.