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

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