Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transformations of the 2-component BKP tau functions

Mengyao Chen, Jipeng Cheng, Jinbiao Wang

TL;DR

The paper develops a Shiota-based Lax formulation for the 2-BKP hierarchy with two differential operators and a mixed operator $H=\partial_1\partial_2+\rho$, and establishes the $\rho$–tau relationship $\rho=2\partial_1\partial_2\log\tau$. It proves that multiplying a 2-BKP tau function by an eigenfunction $q$ yields another tau function, with explicit Darboux updates of the Lax triple and a squared eigenfunction potential $\Omega$ governing transformed eigenfunctions; this underpins a broad Darboux-transformation framework for the hierarchy. The paper then analyzes the $(M_1,M_2)$-reduction, showing how these tau-transformations interact with the reduced Lax structure and the associated bilinear form. Finally, it connects to additional symmetries and Pfaffian identities, deriving Fay-type relations and Pfaffian expressions for iterated tau-transformations, and highlighting links to DKP/Drinfeld–Sokolov-type reductions.

Abstract

The 2-component BKP (2-BKP) hierarchy is an important integrable system corresponding to the infinite dimensional Lie algebras $b_{\infty}$ and $d_{\infty}$, which contains Novikov-Veselov equation and can be used to describe the total descendent potential of D type singularity. Here we firstly introduce the projections of the mixed pseudo-differential operators to rewrite the 2-BKP Lax equation in the Shiota construction, where the scalar Lax operators involving two differential operators $\partial_1$ and $\partial_2$ are used. Based upon this, the $(M_1,M_2)$-reduction of the 2-BKP hierarchy is given. After that, we give the most important result of this paper, i.e., the transformations of the 2-BKP tau functions, which are in fact the 2-BKP Darboux transformations. Here we further give the corresponding changes in the 2-BKP Lax operators. Also the corresponding results are investigated for the reduction case. Finally, the additional symmetries can be viewed as the special cases of the transformations of the 2-BKP tau functions. Besides, we discuss the Pfaffian identities of the 2-BKP tau functions by successive applications of the above transformations, which are closely related with the 2-BKP addition formulae.

Transformations of the 2-component BKP tau functions

TL;DR

The paper develops a Shiota-based Lax formulation for the 2-BKP hierarchy with two differential operators and a mixed operator , and establishes the –tau relationship . It proves that multiplying a 2-BKP tau function by an eigenfunction yields another tau function, with explicit Darboux updates of the Lax triple and a squared eigenfunction potential governing transformed eigenfunctions; this underpins a broad Darboux-transformation framework for the hierarchy. The paper then analyzes the -reduction, showing how these tau-transformations interact with the reduced Lax structure and the associated bilinear form. Finally, it connects to additional symmetries and Pfaffian identities, deriving Fay-type relations and Pfaffian expressions for iterated tau-transformations, and highlighting links to DKP/Drinfeld–Sokolov-type reductions.

Abstract

The 2-component BKP (2-BKP) hierarchy is an important integrable system corresponding to the infinite dimensional Lie algebras and , which contains Novikov-Veselov equation and can be used to describe the total descendent potential of D type singularity. Here we firstly introduce the projections of the mixed pseudo-differential operators to rewrite the 2-BKP Lax equation in the Shiota construction, where the scalar Lax operators involving two differential operators and are used. Based upon this, the -reduction of the 2-BKP hierarchy is given. After that, we give the most important result of this paper, i.e., the transformations of the 2-BKP tau functions, which are in fact the 2-BKP Darboux transformations. Here we further give the corresponding changes in the 2-BKP Lax operators. Also the corresponding results are investigated for the reduction case. Finally, the additional symmetries can be viewed as the special cases of the transformations of the 2-BKP tau functions. Besides, we discuss the Pfaffian identities of the 2-BKP tau functions by successive applications of the above transformations, which are closely related with the 2-BKP addition formulae.

Paper Structure

This paper contains 17 sections, 43 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. The Lax structure of the 2-BKP hierarchy and its reduction
  3. The transformations of 2-BKP tau functions
  4. Applications in the additional symmetries and Pfaffian identities
  5. The Lax equation of the 2-BKP hierarchy
  6. Projections of the mixed pseudo-differential operators
  7. From the bilinear equation to the Lax equation
  8. From the Lax equation to the bilinear equation
  9. The $(M_1,M_2)$-reduction of the 2-BKP hierarchy
  10. Transformations of the 2-BKP tau functions
  11. The squared eigenfunction potentials
  12. Proof of Theorem \ref{['mainqtau']}
  13. Transformations in the reduction case
  14. Additional symmetries and Pfaffian identities of the 2-BKP hierarchy
  15. Additional symmetries of the 2-BKP hierarchy
  16. ...and 2 more sections

Key Result

Theorem 1

Given the tau function $\tau(\mathbf{t})$ and the eigenfunction $q(\mathbf{t})$ of the 2-BKP hierarchy, is another 2-BKP tau function, i.e., satisfying the 2-BKP bilinear equation (2bkptaubilinear). Further if $(L_1,L_2,H)$ is the 2-BKP Lax triple corresponding to $\tau(\mathbf{t})$ by (laxexpression) and (rho), then the new tau function $\tau^{[1]}(\mathbf{t})$ will be corresponding to the new 2

Theorems & Definitions (87)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Corollary 1
  • Remark 3
  • Proposition 1
  • Corollary 2
  • Lemma 1
  • Theorem 2
  • Remark 4
  • ...and 77 more