Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2604.16998 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911642553417728 |
|---|---|
| author | Athanassoulis, Agissilaos |
| author_facet | Athanassoulis, Agissilaos |
| contents | The Alber equation is the mixed-state nonlinear Schrödinger equation with singular ($δ$-interaction) kernel. It is used in the modeling of stochastic ocean waves, where it appears with the focusing sign in the nonlinearity, on $d=1.$ The main result of the paper is global well-posedness for self-adjoint, non-negative data in the Schatten-Sobolev space $H^1\mathfrak{S}^1(\mathbb{T}),$ for both the focusing and defocusing cases. The Schatten class norms achieve control of the position density without derivative loss, and a systematic Fourier-Galerkin argument tailored to the $δ$ kernel allows us to establish several qualitative properties of the solution, including energy conservation. In the focusing case, Hoffmann-Ostenhof and Gagliardo-Nirenberg estimates yield a global a priori $H^1\mathfrak{S}^1$ bound with no smallness condition. Non-negativity is a structural requirement for the energy argument to work. The propagation of higher Sobolev regularity $H^s\mathfrak{S}^1$ follows. As an application, small perturbations around Penrose-stable backgrounds are shown to grow at most polynomially in $H^1\mathfrak{S}$ over long timescales. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16998 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global solutions for the Alber equation in $H^1\mathfrak{S}^1(\mathbb{T})$ Athanassoulis, Agissilaos Analysis of PDEs 35Q55, 35B35, 47B10, 37A60 The Alber equation is the mixed-state nonlinear Schrödinger equation with singular ($δ$-interaction) kernel. It is used in the modeling of stochastic ocean waves, where it appears with the focusing sign in the nonlinearity, on $d=1.$ The main result of the paper is global well-posedness for self-adjoint, non-negative data in the Schatten-Sobolev space $H^1\mathfrak{S}^1(\mathbb{T}),$ for both the focusing and defocusing cases. The Schatten class norms achieve control of the position density without derivative loss, and a systematic Fourier-Galerkin argument tailored to the $δ$ kernel allows us to establish several qualitative properties of the solution, including energy conservation. In the focusing case, Hoffmann-Ostenhof and Gagliardo-Nirenberg estimates yield a global a priori $H^1\mathfrak{S}^1$ bound with no smallness condition. Non-negativity is a structural requirement for the energy argument to work. The propagation of higher Sobolev regularity $H^s\mathfrak{S}^1$ follows. As an application, small perturbations around Penrose-stable backgrounds are shown to grow at most polynomially in $H^1\mathfrak{S}$ over long timescales. |
| title | Global solutions for the Alber equation in $H^1\mathfrak{S}^1(\mathbb{T})$ |
| topic | Analysis of PDEs 35Q55, 35B35, 47B10, 37A60 |
| url | https://arxiv.org/abs/2604.16998 |