The direct linearization scheme with the Lam\'e function: The KP equation and reductions

TL;DR

This paper develops an elliptic direct linearization scheme involving Lamé functions for the KP equation and its reductions, enabling the construction of elliptic soliton solutions and analysis of their interactions on periodic backgrounds.

Contribution

It introduces a novel elliptic DL scheme with integral equations and soliton formulas, extending the direct linearization method to elliptic functions and solutions.

Findings

Constructed elliptic soliton solutions confirmed by Lax pairs.

Developed a Marchenko equation for elliptic solitons.

Demonstrated soliton interactions on periodic backgrounds.

Abstract

The paper starts from establishing an elliptic direct linearization (DL) scheme for the Kadomtsev-Petviashvili equation. The scheme consists of an integral equation (involving the Lam\'e function) and a formula for elliptic soliton solutions, which can be confirmed by checking Lax pair. Based on analysis of real-valuedness of the Weierstrass functions, we are able to construct a Marchenko equation for elliptic solitons. A mechanism to obtain nonsingular real solutions from this elliptic DL scheme is formulated. By utilizing elliptic $N$th roots of unity and reductions, the elliptic DL schemes, Marchenko equations and nonsingular real solutions are studied for the Korteweg-de Vries equation and Boussinesq equation. Illustrations of the obtained solutions show solitons and their interactions on a periodic background.