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arXiv · 2501.06476

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

Xing Li·Ying-ying Sun·Da-jun Zhang
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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.

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Xing Li, Ying-ying Sun, Da-jun Zhang. 2025-01-11. The direct linearization scheme with the Lam\'e function: The KP equation and reductions. https://arxiv.org/abs/2501.06476

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