Elliptic (N,N^\prime)-Soliton Solutions of the lattice KP Equation

TL;DR

This paper constructs elliptic soliton solutions for various integrable lattice KP equations using an elliptic Cauchy kernel, extending to continuum limits and reductions to KdV-type equations.

Contribution

It introduces a novel method to generate elliptic soliton solutions for lattice KP equations and their reductions, broadening the understanding of elliptic integrable systems.

Findings

Constructed elliptic soliton solutions for lattice KP, mKP, SKP, and Hirota's bilinear KP equations.

Extended solutions to continuum limits of these equations.

Discussed reduction to elliptic soliton solutions of KdV-type lattice equations.

Abstract

Elliptic soliton solutions, i.e., a hierarchy of functions based on an elliptic seed solution, are constructed using an elliptic Cauchy kernel, for integrable lattice equations of Kadomtsev-Petviashvili (KP) type. This comprises the lattice KP, modified KP (mKP) and Schwarzian KP (SKP) equations as well as Hirota's bilinear KP equation, and their successive continuum limits. The reduction to the elliptic soliton solutions of KdV type lattice equations is also discussed.