Tau-Functions and Generalized Integrable Hierarchies

TL;DR

This paper develops a tau-function formalism for generalized zero-curvature integrable hierarchies, including Drinfel'd-Sokolov hierarchies, clarifying their relation to Hirota-type hierarchies and establishing a direct link between variables.

Contribution

It introduces a tau-function approach for a broad class of integrable hierarchies, unifying different formalisms and clarifying their interconnections.

Findings

Constructed the tau-function formalism for generalized hierarchies.

Established the relation between zero-curvature variables and tau-functions.

Clarified the connection between zero-curvature and Hirota-type hierarchies.

Abstract

The tau-function formalism for a class of generalized ``zero-curvature'' integrable hierarchies of partial differential equations, is constructed. The class includes the Drinfel'd-Sokolov hierarchies. A direct relation between the variables of the zero-curvature formalism and the tau-functions is established. The formalism also clarifies the connection between the zero-curvature hierarchies and the Hirota-type hierarchies of Kac and Wakimoto.