We consider the Calogero-Sutherland derivative nonlinear Schr\"odinger 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 $H^s_+(\mathbb{ T}),$ $s\geq0\,.$
Categories: math.AP