On the Integrable Generalization of the 1D Toda Lattice

TL;DR

This paper introduces a generalized 1D Toda Lattice, derives its Lax representation, and explores solutions and relations to other integrable systems, expanding understanding of nonlinear lattice models.

Contribution

It presents a new integrable generalization of the Toda Lattice, including its Lax pair, explicit solutions, and connections to the nonlinear Schrödinger and Heisenberg ferromagnetic equations.

Findings

Lax representation for the generalized Toda Lattice is derived.

Explicit tau-function solutions are constructed for N=3.

Connections to nonlinear Schrödinger and Heisenberg ferromagnetic equations are established.

Abstract

A generalized Toda Lattice equation is considered. The associated linear problem (Lax representation) is found. For simple case N=3 the $\tau$-function Hirota form is presented that allows to construct an exast solutions of the equations of the 1DGTL. The corresponding hierarchy and its relations with the nonlinear Schrodinger equation and Hersenberg ferromagnetic equation are discussed.