Integrable discretizations for lattice systems: local equations of motion and their Hamiltonian properties

Abstract

We develop the approach to the problem of integrable discretization based on the notion of $r$--matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying continuous time systems. A common feature of the discretizations obtained in this approach is non--locality. We demonstrate how to overcome this drawback. Namely, we introduce the notion of localizing changes of variables and construct such changes of variables for a large number of examples, including the Toda and the relativistic Toda lattices, the Volterra lattice and its integrable perturbation, the second flows of the Toda and of the Volterra hierarchies, the modified Volterra lattice, the Belov-Chaltikian lattice, the Bogoyavlensky lattices, the Bruschi-Ragnisco lattice. We also introduce a novel class of constrained lattice KP systems, discretize all of them, and find the corresponding localizing change of variables. Pulling back the differential equations of motion under the localizing changes of variables, we find also (sometimes novel) integrable one-parameter perturbations of integrable lattice systems. Poisson properties of the localizing changes of variables are also studied: they produce interesting one-parameter deformations of the known Poisson algebras.