Applications of the Sylvester equation for the lattice BKP system

The Sylvester equation plays an important role in many branches of mathematical physics. The goal of this paper is to show that a special case of the Sylvester equation can be related to the lattice B-type Kadomtsev–Petviashvili (BKP) system via the generalized Cauchy matrix method. We use the variables given in the Sylvester equation to define the $\tau$ function and several scalar functions that are closely related to the lattice BKP equation. After rederiving the lattice BKP equation, we make it clear that besides its multisoliton solutions, various other types of exact solutions also exist. Furthermore, Lax pairs for the lattice BKP equation are obtained in different ways.