Asymptotic stability of solitons for near-cubic NLS equation with an internal mode
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913536543817728 |
|---|---|
| author | Rialland, Guillaume |
| author_facet | Rialland, Guillaume |
| contents | We consider perturbations of the one-dimensional cubic Schrödinger equation, of the form $i \, \partial_t ψ+ \partial_x^2 ψ+ |ψ|^2 ψ+ g( |ψ|^2 ) ψ= 0$. Under hypotheses on the function $g$ that can be easily verified in some cases (such as $g(s) = s^σ$ with $σ>1$), we show that the linearized problem around a small solitary wave presents a unique internal mode. Moreover, under an additional hypothesis (the Fermi golden rule) that can also be verified in the case of powers $g(s) = s^σ$, we prove the asymptotic stability of the solitary waves with small frequencies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13980 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic stability of solitons for near-cubic NLS equation with an internal mode Rialland, Guillaume Analysis of PDEs We consider perturbations of the one-dimensional cubic Schrödinger equation, of the form $i \, \partial_t ψ+ \partial_x^2 ψ+ |ψ|^2 ψ+ g( |ψ|^2 ) ψ= 0$. Under hypotheses on the function $g$ that can be easily verified in some cases (such as $g(s) = s^σ$ with $σ>1$), we show that the linearized problem around a small solitary wave presents a unique internal mode. Moreover, under an additional hypothesis (the Fermi golden rule) that can also be verified in the case of powers $g(s) = s^σ$, we prove the asymptotic stability of the solitary waves with small frequencies. |
| title | Asymptotic stability of solitons for near-cubic NLS equation with an internal mode |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.13980 |