The direct linearization scheme with the Lamé function: The KP equation and reductions
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916948221100032 |
|---|---|
| author | Li, Xing Sun, Ying-ying Zhang, Da-jun |
| author_facet | Li, Xing Sun, Ying-ying Zhang, Da-jun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06476 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The direct linearization scheme with the Lamé function: The KP equation and reductions Li, Xing Sun, Ying-ying Zhang, Da-jun Exactly Solvable and Integrable Systems 33E05, 35C08, 35Q51, 37K40 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. |
| title | The direct linearization scheme with the Lamé function: The KP equation and reductions |
| topic | Exactly Solvable and Integrable Systems 33E05, 35C08, 35Q51, 37K40 |
| url | https://arxiv.org/abs/2501.06476 |