Toda lattice realization of integrable hierarchies

TL;DR

This paper demonstrates that scalar integrable hierarchies can be embedded within the Toda lattice hierarchy, providing a new realization that parallels the generality of the Drinfeld--Sokolov approach.

Contribution

It introduces a novel realization of scalar integrable hierarchies through the Toda lattice hierarchy, expanding the understanding of their structural relationships.

Findings

Integrable hierarchies can be embedded in the Toda lattice hierarchy.

The realization is as general as the Drinfeld--Sokolov approach.

Multiple examples illustrate the embedding process.

Abstract

We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudodifferential Lax operator, can be embedded in the Toda lattice hierarchy. Such a realization in terms the Toda lattice hierarchy seems to be as general as the Drinfeld--Sokolov realization.