## Sharp well-posedness results for the Schrödinger-Benjamin-Ono system.(English)Zbl 1342.35329

The author studies the coupled Schrödinger-Benjamin-Ono system \begin{aligned} i\partial_{t}u+\partial_{x}^{2}u & =\alpha u v, \\ \partial_{t}v+\nu\mathcal{H}\partial_{x}^{2}v & =\beta\partial_{x}(|u|^{2}).\end{aligned} Here, $$\mathcal{H}$$ is the Hilbert transform, and the independent variable $$x$$ is taken to be in $$\mathbb{R}$$. The non-resonant case is the case in which $$|\nu|\neq 1$$. Prior existence results have been established, showing well-posedness of this system in the Sobolev spaces $$H^{s}\times H^{s-1/2}$$.
Here, the author studies well-posedness in $$H^{s}\times H^{s'}$$, without requiring $$s'=s-1/2.$$ Under certain conditions on $$s$$, $$s'$$, local well-posedness is proved. Furthermore, in both the resonant and non-resonant cases, for a variety of values of $$s$$, $$s'$$, the failure of the solution map to be $$C^{2}$$ is proved.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35G25 Initial value problems for nonlinear higher-order PDEs
Full Text: