We consider a type of nonlocal nonlinear derivative Schrödinger equation on the torus, called the Calogero-Sutherland DNLS equation.
We derive an explicit formula to the solution of this nonlinear PDE. Moreover, using the integrability tools, we establish the global well-posedness of this equation in all the Hardy-Sobolev spaces $H^s_+(\mathbb{T})$, $s\geq 0$, down to the critical regularity space, and under a mass assumption on the initial data for the focusing equation, and for arbitrary initial data for the defocusing equation.
Finally, a sketch of the proof for extending the flow to the critical regularity $L^2_+(\mathbb{T})$ will be presented.
