Abstract

We consider the Calogero–Sutherland derivative nonlinear Schrödinger equation in the focusing (with sign $+$) and defocusing case (with sign $-$)

where $\mathrm{\Pi }$ is the Szegő projector $\mathrm{\Pi }\left({\sum <!--FUNCTION\; APPLICATION-->}_{n\in \mathbb{Z}}û(n){e<!--FUNCTION\; APPLICATION-->}^{inx}\right)={\sum <!--FUNCTION\; APPLICATION-->}_{n\ge 0}û(n){e<!--FUNCTION\; APPLICATION-->}^{inx}$. 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 $H_{+}^s(\mathbb{𝕋})$, $s\ge 0$.

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