Additional symmetries of the KP-mKP hierarchy and Virasoro constraints to the Burgers-KdV hierarchy

TL;DR

This paper explores the symmetries of the KP-mKP hierarchy, constructs Fay identities, and demonstrates how these symmetries lead to Virasoro constraints for the Burgers-KdV hierarchy and its extensions.

Contribution

It introduces new differential Fay identities and constructs additional symmetries for the KP-mKP hierarchy, linking them to Virasoro constraints in a novel way.

Findings

Derived a class of differential Fay identities.

Constructed additional symmetries as linear actions on tau functions.

Reproved Virasoro constraints for Burgers-KdV hierarchy and extended to higher orders.

Abstract

A KP-mKP hierarchy was introduced recently via pseudo-differential operators containing two derivations. In this paper, for the KP-mKP hierarchy we derive a class of (differential) Fay identities and construct a series of additional symmetries. Moreover, the additional symmetries are represented as certain linear actions on the tau functions of the hierarchy, with the help of the Adler-Shiota-van Moerbeke formula. As an application, we reprove the Virasoro constraints to the tau functions of the Burgers-KdV hierarchy, and such results are generalized to its higher order extensions regarded as reductions of the KP-mKP hierarchy.