Integrable variants of the Leznov lattice and the differential-difference KP equations

Abstract

By introducing trigonometric-type bilinear operators, we propose two novel discrete integrable equations that can be viewed as variants of the Leznov lattice and the differential-difference Kadomtsev–Petviashvili (D$\Delta$KP) equations. It turns out that both equations admit various solutions, including general Grammian determinant, soliton, lump, rogue wave, and breather solutions, which are expressed by explicit and closed forms. Moreover, $g$-periodic wave solutions are also constructed in terms of Riemann theta function. Numerical three-periodic wave solutions are successfully computed by using a deep neural network. Finally, we construct a continuum limit, through which we reveal clear links between the variant D$\Delta$KP equation and the Kadomtsev–Petviashvili-I (KPI) equation from both the equation and solution perspectives.