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Nonlinear Sciences > Exactly Solvable and Integrable Systems

arXiv:2501.06476 (nlin)
[Submitted on 11 Jan 2025 (v1), last revised 18 Jan 2025 (this version, v2)]

Title:The direct linearization scheme with the Lamé function: The KP equation and reductions

Authors: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 Lamé 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.
Comments: 26 pages
Subjects: Exactly Solvable and Integrable Systems (nlin.SI)
MSC classes: 33E05, 35C08, 35Q51, 37K40
Cite as: arXiv:2501.06476 [nlin.SI]
  (or arXiv:2501.06476v2 [nlin.SI] for this version)
  https://doi.org/10.48550/arXiv.2501.06476

Submission history

From: Da-jun Zhang [view email]
[v1] Sat, 11 Jan 2025 08:07:26 UTC (623 KB)
[v2] Sat, 18 Jan 2025 04:51:52 UTC (624 KB)
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