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

Zongyao Feng, Lumin Geng, Chao-Zhong Wu

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.

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

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.
Paper Structure (9 sections, 18 theorems, 128 equations)

This paper contains 9 sections, 18 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Pseudo-differential operators and the KP-mKP hierarchy
  3. Pseudo-differential operators with two derivations
  4. The KP-mKP hierarchy
  5. Identities of Fay type and their applications
  6. Additional symmetries
  7. The Adler-Shiota-van Moerbeke formula
  8. Virasoro constraints to the Burgers-KdV hierarchy and its higher order extensions
  9. Concluding remarks

Key Result

Lemma 2.1

For any pseudo-differential operators $F, G\in\mathcal{E}$ that contain only powers in $\partial_\nu$ with $\nu\in\{1,2\}$, the following equality holds true:

Theorems & Definitions (24)

  • Lemma 2.1: see, for example, DKJM-KPBKP
  • Theorem 2.2: GHW2024
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Corollary 3.5
  • Lemma 3.6
  • Lemma 3.7
  • proof
  • ...and 14 more