Discrete time Toda systems

TL;DR

This paper explores key concepts in discrete integrable systems, illustrating them through discrete time Toda lattices and their relativistic versions, covering topics like Bäcklund transformations, multi-Hamiltonian structures, and multi-dimensional consistency.

Contribution

It provides a comprehensive overview of modern discrete integrable system concepts, specifically applying them to discrete time Toda lattices and related models.

Findings

Discrete time Toda lattices exemplify integrability concepts.

Multi-dimensional consistency characterizes the integrability of these systems.

Connections between quad-equations and Laplace-type systems are established.

Abstract

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.