Multi-component Toda lattice hierarchy

TL;DR

This paper extends the Toda lattice hierarchy to multiple components, deriving its formalism, tau-function, and solutions, and embedding it into the universal hierarchy using fermion techniques.

Contribution

It introduces a multi-component Toda lattice hierarchy with N discrete variables, extending previous models and connecting it to the multi-component KP hierarchy.

Findings

Derived the Lax formalism for the multi-component hierarchy

Proved the existence of the tau-function for the hierarchy

Constructed explicit multi-component soliton solutions

Abstract

We give a detailed account of the N -component Toda lattice hierarchy. This hierarchy is an extended version of the one introduced by Ueno and Takasaki. Our version contains N discrete variables rather than one. We start from the Lax formalism, deduce the bilinear relation for the wave functions from it and then, based on the latter, prove existence of the tau-function. We also show how the multi-component Toda lattice hierarchy is embedded into the universal hierarchy which is basically the multi-component KP hierarchy. At last, we show how the bilinear integral equation for the tau-function can be obtained using the free fermion technique. An example of exact solutions (a multi-component analogue of one-soliton solutions) is given.