Special decompositions and linear superpositions of nonlinear systems: BKP and dispersionless BKP equations

The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is analyzed. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

Intersection numbers on $$ {\overline{M}}_{g,n} $$ and BKP hierarchy

In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.

KdV solves BKP

In this note, we prove that any τ-function of the Korteweg–de Vries (KdV) hierarchy also solves the type B Kadomtsev–Petviashvili (BKP) hierarchy after a simple rescaling of times.

Some results of the BKP hierarchy as the Kupershmidt reduction of the modified KP hierarchy

In this paper, the BKP hierarchy is viewed as the Kupershmidt reduction of the modified KP hierarchy. Then based upon this fact, the gauge transformation of the BKP hierarchy are obtained again from the corresponding results of the modified KP hierarchy. Also the constrained BKP hierarchy is constructed from the constrained modified KP hierarchy, and the corresponding gauge transformations are investigated. Particularly, it is found that there is a new kind of gauge transformations generated by the wave functions in the constrained BKP hierarchy.

The 2-Component BKP Grassmannian and Simple Singularities of Type D

It was proved in 2010 that the principal Kac–Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac–Wakimoto hierarchy. We find explicitly the point in the Grassmannian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$.

Darboux transformations of new supersymmetric CKP and BKP hierarchies

In this paper, we construct a supersymmetric CKP(SCKP) hierarchy and its Darboux transformations. These Darboux transformations can generate new solutions from seed solutions by using supersymmetric functions. Further, we obtain the supersymmetric Sawada–Kotera equation from the reduction of supersymmetric CKP hierarchy. After that, using a new B type condition, we also construct a new supersymmetric BKP(SBKP) hierarchy which is different from the supersymmetric BKP hierarchy defined by Ramos and Stanciu. Meanwhile, the Darboux transformations of this new SBKP hierarchy are derived.