On an elliptic extension of the Kadomtsev–Petviashvili equation

A generalization of the lattice potential Kadomtsev–Petviashvili (LPKP) equation is presented, using the method of direct linearization based on an elliptic Cauchy kernel. This yields a 3+1-dimensional lattice system with one of the lattice shifts singled out. The integrability of the lattice system is considered, presenting a Lax representation and soliton solutions. An associated continuous system is also derived, yielding a 3+1-dimensional generalization of the potential KP equation associated with an elliptic curve.