The Modified Toda Hierarchy

TL;DR

This paper introduces the mToda hierarchy, a new integrable hierarchy generalizing the Toda hierarchy within Sato theory, establishing its equivalence with Lax formulations, Miura links, and Darboux transformations.

Contribution

It constructs the mToda hierarchy from bilinear equations of the 2-component modified KP hierarchy and demonstrates its connections to Toda hierarchy via Miura links.

Findings

Established the equivalence of mToda hierarchy with Lax formulations.

Demonstrated Miura links between Toda and mToda hierarchies.

Constructed Darboux transformations for both hierarchies.

Abstract

In this paper, modified Toda (mToda) equation is generalized to form an integrable hierarchy in the framework of Sato theory, which is therefore called mToda hierarchy. Inspired by the fact that Toda hierarchy is 2-component generalization of usual KP hierarchy, mToda hierarchy is constructed from bilinear equations of 2-component first modified KP hierarchy, where we provide the corresponding equivalence with Lax formulations. Then it is demonstrated that there are Miura links between Toda and mToda hierarchies, which means the definition of mToda hierarchy here is reasonable. Finally, Darboux transformations of the Toda and mToda hierarchies are also constructed by using the aforementioned Miura links.