Tau function and Hirota bilinear equations for the Extended bigraded Toda Hierarchy

TL;DR

This paper extends the Sato theory to the extended bigraded Toda hierarchy, establishing tau functions, Hirota bilinear identities, and Fay-like identities, thus advancing the mathematical framework of this integrable system.

Contribution

It generalizes the Sato theory to the EBTH, introducing new Hirota bilinear equations and tau functions with operator-valued coefficients.

Findings

Established the existence of tau functions for EBTH

Derived Hirota bilinear identities and Fay-like identities

Proved the validity of Hirota bilinear equations involving differential operators

Abstract

In this paper we generalize the Sato theory to the extended bigraded Toda hierarchy (EBTH). We revise the definition of the Lax equations,give the Sato equations, wave operators, Hirota bilinear identities (HBI) and show the existence of $tau$ function $\tau(t)$. Meanwhile we prove the validity of its Fay-like identities and Hirota bilinear equations (HBEs) in terms of vertex operators whose coefficients take values in the algebra of differential operators. In contrast with HBEs of the usual integrable system, the current HBEs are equations of product of operators involving $e^{\partial_x}$ and $\tau(t)$.