Transformations of the 2-component BKP tau functions

TL;DR

This paper introduces transformations of 2-component BKP tau functions, including Darboux transformations and their effects on Lax operators, with implications for reductions, symmetries, and Pfaffian identities in integrable systems.

Contribution

It presents the first detailed study of Darboux transformations for 2-BKP tau functions and explores their impact on Lax operators, reductions, and identities.

Findings

Derived explicit Darboux transformations for 2-BKP tau functions

Connected transformations with additional symmetries and Pfaffian identities

Provided new insights into the structure of 2-BKP hierarchy and its 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.