On a coupled Kadomtsev--Petviashvili system associated with an elliptic curve

TL;DR

This paper reexamines an elliptic curve-associated coupled Kadomtsev--Petviashvili system using direct linearisation, leading to new insights, a Lax pair, multi-soliton solutions, and reductions to related integrable systems.

Contribution

It introduces a Lax pair and multi-soliton solutions for the elliptic coupled KP system, and explores reductions and new solution classes within the integrability framework.

Findings

Constructed a Lax pair for the elliptic coupled KP system.

Derived multi-soliton solutions parametrized by elliptic curve points.

Discussed reductions to elliptic coupled KdV and Boussinesq systems.

Abstract

The coupled Kadomtsev--Petviashvili system associated with an elliptic curve, proposed by Date, Jimbo and Miwa [J. Phys. Soc. Jpn., 52:766--771, 1983], is reinvestigated within the direct linearisation framework, which provides us with more insights into the integrability of this elliptic model from the perspective of a general linear integral equation. As a result, we successfully construct for the elliptic coupled Kadomtsev--Petviashvili system not only a Lax pair composed of differential operators in $2\times2$ matrix form but also multi-soliton solutions with phases parametrised by points on the elliptic curve. Dimensional reductions based on the direct linearisation, to the elliptic coupled Korteweg-de Vries and Boussinesq systems, are also discussed. In addition, a novel class of solutions are obtained for the $D_\infty$-type Kadomtsev--Petviashvili equation with nonzero constant background as a byproduct.