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

Abstract

We consider the Calogero-Sutherland derivative nonlinear Schrödinger equation in the focusing (with sign $+$) and defocusing case (with sign $-$) $$i{\partial }_{t}u+{\partial }_x^{2}u\pm \frac{2}{i}{\partial }_x\Pi (|u{|}^2)u=0,(t,x)\in \mathbb{R}\times \mathbb{T},$$ where $\Pi$ is the Szegő projector $\Pi \left({\sum }_{n\in \mathbb{Z}}\hat{u}(n){\mathrm{e}}^{inx}\right)={\sum }_{n\ge 0}\hat{u}(n){\mathrm{e}}^{inx}$. 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 $H_{+}^s(\mathbb{T}),$ $s\ge 0,$ 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 $H_{+}^s(\mathbb{T}),$ $s\ge 0.$

38 pages. Various typos were corrected. Comments are welcome !

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• Integrable hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation
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• On the well-posedness problem for the derivative nonlinear Schrödinger equation
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