On the global well-posedness of theCalogero–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 -)where Π is the Szegő projector Π n∈Z u(n) e inx = n≥0 u(n) 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 + (T) , s ≥ 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 + (T) , s ≥ 0 .

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