Transformations of the 2-component BKP tau functions

Associated with the infinite-dimensional Lie algebras $b_{\infty}$ and $d_{\infty}$, the two-component BKP (2-BKP) hierarchy constitutes a significant class of integrable systems. This hierarchy accommodates the Novikov–Veselov equation and admits a concrete realization in the characterization of the total descendant potential for D-type singularities. In this paper, we reformulate the Lax equation of the 2-BKP hierarchy within Shiota’s theoretical framework by introducing projection operations on mixed pseudo-differential operators, where scalar Lax operators parameterized by two independent differential operators $\partial_1$ and $\partial_2$ are adopted. Building on this, we establish the explicit $(M_1,M_2)$-reduction of the 2-BKP hierarchy. We then systematically construct tau function transformations for the 2-BKP hierarchy and demonstrate that such transformations exactly correspond to the Darboux transformations of the 2-BKP system, with the induced variations of the associated Lax operators explicitly derived. Moreover, we extend these transformation results to the $(M_1,M_2)$-reduction. We further prove that the additional symmetries of the 2-BKP hierarchy can be recovered as particular instances of the proposed tau function transformations. By iteratively implementing these Darboux-type transformations, we derive a series of Pfaffian identities for 2-BKP tau functions, which serve as generalized extensions of the classical addition formulae existing in previous references.