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Pré-Publication, Document De Travail Année : 2023

On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schrödinger equation

Résumé

We consider the Calogero-Sutherland derivative nonlinear Schrödinger equation in the focusing (with sign $+$) and defocusing case (with sign $-$) $$ i\partial_tu+\partial_x^2u\,\pm\,\frac2i\,\partial_x\Pi(|u|^2)u=0\,,\qquad (t,x)\in\mathbb{R}\times\mathbb{T}, $$ where $\Pi$ is the Szeg\H{o} projector $\Pi\left(\sum_{n\in \mathbb{Z}}\widehat{u}(n)\mathrm{e}^{inx}\right)=\sum_{n\geq 0 }\widehat{u}(n)\mathrm{e}^{inx}$. 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 $H^s_+(\mathbb{T}),$ $s\geq0\,.$
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Dates et versions

hal-04007823 , version 1 (28-02-2023)
hal-04007823 , version 2 (04-12-2023)

Identifiants

  • HAL Id : hal-04007823 , version 1

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Rana Badreddine. On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schrödinger equation. 2023. ⟨hal-04007823v1⟩
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