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.In what follows we suppose, for the sake of notational simplicity, that G is a matrix group, and write the adjoint action of the group on g as a conjugation by the corresponding matrices. Correspondingly, we shall call Ad-covariant functions f : g → g also "conjugation covariant". This notation has an additional advantage of being applicable also to functions F : g → G.Namely, we shall call such a function conjugation covariant, if F (Ad g · T ) = gF (T )g −1 .Theorem 3.1. Let f : g → g be a conjugation covariant function. Then the solution of the differential equation (3.3) with the initial condition T (0) = T 0 is given, at least for sufficiently small t, by T (t) = Π −1 + e tf (T 0 ) · T 0 · Π + e tf (T 0 ) = Π − e tf (T 0 ) · T 0 · Π −1 − e tf (T 0 ) .(3.5)

A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions

“…where x n = x(n, t) and p n = p(n, t) are the real functions of discrete variable n and time variable t. When α → 0, Equation (1) reduces to the famous TL system as a very remarkable generalization of TL system; thus, the study of Equation ( 1) is of great significance for understanding the physical phenomena described by the TL system. Equation (1) can also be considered as a one-parameter perturbation of TL system, and in a certain physical interpretation, the small parameter α has the meaning of the inverse speed of light [10,11].…”

“…In Ref. [11], Suris proposed some discrete time Toda systems including the RTL − (α) system (see System (8.2) in Ref. [11]).…”

“…[11], Suris proposed some discrete time Toda systems including the RTL − (α) system (see System (8.2) in Ref. [11]). Although Equation (1) has the same abbreviation as system (8.2) in the literature [11], by careful comparison, we found that they are different from each other.…”

“…[11]). Although Equation (1) has the same abbreviation as system (8.2) in the literature [11], by careful comparison, we found that they are different from each other. Here, we need to point out that the RTL − (α) system that we study here is different from the RTL equation in Refs.…”

Nekhoroshev Stability for the Dirichlet Toda Lattice

“…Moreover, the Toda lattice is a model with has numerous applications in the physical sciences, ranging from solid state physics [5] and quantum field theory [6] to DNA transcription [7], to mention just a few. Some review articles on the history of the research on the Toda lattice have been published very recently [8][9][10].…”