On an elliptic extension of the Kadomtsev-Petviashvili equation

TL;DR

This paper introduces an elliptic extension of the KP equation, deriving a new (3+1)-dimensional lattice system and its continuous counterpart, demonstrating integrability through Lax pairs and soliton solutions.

Contribution

It presents a novel elliptic generalization of the KP equation using direct linearisation, resulting in a new lattice system and its continuous form with proven integrability.

Findings

Derived a (3+1)-dimensional lattice system with elliptic structure.

Established Lax representation confirming integrability.

Constructed explicit soliton solutions for the system.

Abstract

A generalisation of the Lattice Potential Kadomtsev-Petviashvili (LPKP) equation is presented, using the method of Direct Linearisation 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 generalisation of the potential KP equation associated with an elliptic curve.