Bi-Hamiltonian structures of Toda type systems

TL;DR

This paper introduces a new family of integrable systems similar to the Toda lattice, providing Lax pairs, Poisson structures, and bi-Hamiltonian formulations, bridging classical Toda and Kostant-Toda systems.

Contribution

It constructs Lax pairs, Poisson structures, and bi-Hamiltonian frameworks for these new systems, expanding the understanding of Toda-type integrable models.

Findings

Established Lax pair representations for the systems

Derived Poisson and bi-Hamiltonian structures

Presented master symmetries for low-dimensional cases

Abstract

In this paper we define a family of systems which have similarities with the Toda lattice. We construct two Lax pair representations and the associate Poisson structures for these systems. These systems lie between the classical Toda lattice and the full Kostant-Toda lattice. A Hamiltonian and a bi-Hamiltonian structure for these systems is constructed and also master symmetries for some low dimensional cases are presented.