On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schrödinger equation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910453282635776 |
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| author | Badreddine, Rana |
| author_facet | Badreddine, Rana |
| contents | We consider the Calogero-Sutherland derivative nonlinear Schrödinger equation in the focusing (with sign $+$) and defocusing case (with sign $-$)
$$
i\partial_tu+\partial_x^2u\,\pm\,\frac2i\,\partial_xΠ(|u|^2)u=0\,,\qquad (t,x)\in\mathbb{R}\times\mathbb{T},
$$ where $Π$ is the Szegő projector $Π\left(\sum_{n\in \mathbb{Z}}\widehat{u}(n)\mathrm{e}^{inx}\right)=\sum_{n\geq 0 }\widehat{u}(n)\mathrm{e}^{inx}$. Thanks to a Lax pair formulation, we derive the explicit solution to this equation. Furthermore, we prove the global well-posedness for this $L^2$-critical equation in all the Hardy Sobolev spaces $H^s_+(\mathbb{T}),$ $s\geq0\,,$ with small $L^2$-initial data in the focusing case, and for arbitrarily $L^2$-data in the defocusing case. In addition, we establish the relative compactness of the trajectories in all $H^s_+(\mathbb{T}),$ $s\geq0\,.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_01087 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schrödinger equation Badreddine, Rana Analysis of PDEs 37K10, 35Q55, 47B35 We consider the Calogero-Sutherland derivative nonlinear Schrödinger equation in the focusing (with sign $+$) and defocusing case (with sign $-$) $$ i\partial_tu+\partial_x^2u\,\pm\,\frac2i\,\partial_xΠ(|u|^2)u=0\,,\qquad (t,x)\in\mathbb{R}\times\mathbb{T}, $$ where $Π$ is the Szegő projector $Π\left(\sum_{n\in \mathbb{Z}}\widehat{u}(n)\mathrm{e}^{inx}\right)=\sum_{n\geq 0 }\widehat{u}(n)\mathrm{e}^{inx}$. Thanks to a Lax pair formulation, we derive the explicit solution to this equation. Furthermore, we prove the global well-posedness for this $L^2$-critical equation in all the Hardy Sobolev spaces $H^s_+(\mathbb{T}),$ $s\geq0\,,$ with small $L^2$-initial data in the focusing case, and for arbitrarily $L^2$-data in the defocusing case. In addition, we establish the relative compactness of the trajectories in all $H^s_+(\mathbb{T}),$ $s\geq0\,.$ |
| title | On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schrödinger equation |
| topic | Analysis of PDEs 37K10, 35Q55, 47B35 |
| url | https://arxiv.org/abs/2303.01087 |