On the Lattice Potential KP Equation

TL;DR

This paper develops a method to derive finite genus solutions for the lattice potential KP equation using spectral problems, Darboux transformations, and nonlinearization techniques, also discussing semi-discrete variants.

Contribution

It introduces a novel approach linking spectral problems and Darboux transformations to derive finite genus solutions for the lpKP equation.

Findings

Finite genus solutions for the lpKP equation are derived.

Connections between discrete spectral problems and the lpKP equation are established.

Semi-discrete potential KP equations are analyzed.

Abstract

The paper presents an approach to derive finite genus solutions to the lattice potential Kadomtsev-Petviashvili (lpKP) equation introduced by F.W. Nijhoff, et al. This equation is rederived from compatible conditions of three replicas of the discrete ZS-AKNS spectral problem, which is a Darboux transformation of the continuous ZS-AKNS spectral problem. With the help of these links and by means of the so called nonlinearization technique and Liouville platform, finite genus solutions of the lpKP equation are derived. Semi-discrete potential KP equations with one and two discrete arguments, respectively, are also discussed.