Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2505.15915 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918107980759040 |
|---|---|
| author | Stewart, Gavin |
| author_facet | Stewart, Gavin |
| contents | We consider solutions to the Benjamin-Ono equation
$$\partial_t u - H \partial_x^2 u = -\partial_x(u^2)$$
that are localized in a reference frame moving to the right with constant speed. We show that any such solution that decays at least like $\langle x\rangle^{-1-ε}$ for some $ε> 0$ in a comoving coordinate frame must in fact decay like $\langle x\rangle^{-2}$. In view of the explicit soliton solutions, this decay rate is sharp. Our proof has two main ingredients. The first is microlocal dispersive estimates for the Benjamin-Ono equation in a moving frame, which allow us to prove spatial decay of the solution provided the nonlinearity has sufficient decay. The second is a careful normal form analysis, which allows us to obtain rapid decay of the nonlinearity for a transformed equation assuming only modest decay of the solution. Our arguments are entirely time dependent, and do not require the solution to be an exact traveling wave. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15915 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On spatial decay for coherent states of the Benjamin-Ono equation Stewart, Gavin Analysis of PDEs 76B15 (Primary) 70K45 35C07 (Secondary) We consider solutions to the Benjamin-Ono equation $$\partial_t u - H \partial_x^2 u = -\partial_x(u^2)$$ that are localized in a reference frame moving to the right with constant speed. We show that any such solution that decays at least like $\langle x\rangle^{-1-ε}$ for some $ε> 0$ in a comoving coordinate frame must in fact decay like $\langle x\rangle^{-2}$. In view of the explicit soliton solutions, this decay rate is sharp. Our proof has two main ingredients. The first is microlocal dispersive estimates for the Benjamin-Ono equation in a moving frame, which allow us to prove spatial decay of the solution provided the nonlinearity has sufficient decay. The second is a careful normal form analysis, which allows us to obtain rapid decay of the nonlinearity for a transformed equation assuming only modest decay of the solution. Our arguments are entirely time dependent, and do not require the solution to be an exact traveling wave. |
| title | On spatial decay for coherent states of the Benjamin-Ono equation |
| topic | Analysis of PDEs 76B15 (Primary) 70K45 35C07 (Secondary) |
| url | https://arxiv.org/abs/2505.15915 |