Discrete time Toda systems

文献摘要

In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including: Time discretization based on the notion of B"acklund 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.

In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including:

Time discretization based on the notion of B"acklund 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.