On reductions of the discrete Kadomtsev--Petviashvili-type equations

The reduction by restricting the spectral parameters $k$ and $k^{\mathrm{\prime }}$ on a generic algebraic curve of degree $\mathcal{N}$ is performed for the discrete AKP, BKP and CKP equations, respectively. A variety of two-dimensional discrete integrable systems possessing a more general solution structure arise from the reduction, and in each case a unified formula for generic positive integer $\mathcal{N}\ge 2$ is given to express the corresponding reduced integrable lattice equations. The obtained extended two-dimensional lattice models give rise to many important integrable partial difference equations as special degenerations. Some new integrable lattice models such as the discrete Sawada–Kotera, Kaup–Kupershmidt and Hirota–Satsuma equations in extended form are given as examples within the framework.