Discrete time Toda systems

Abstract

In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including:• Time discretization based on the notion of Bäcklund transformation.• Symplectic realizations of multi-Hamiltonian structures.• Interrelations between discrete 1D systems and lattice 2D systems.• Multi-dimensional consistency as integrability of discrete systems.• Interrelations between integrable systems of quad-equations and integrable systems of Laplace type.• Pluri-Lagrangian structure as integrability of discrete variational systems.All these concepts are illustrated by the discrete time Toda lattices and their relativistic analogs.