Sharp Well-Posedness Results for the Schr\"odinger-Benjamin-Ono System

This work is concerned with the Cauchy problem for a coupled Schr\"odinger-Benjamin-Ono system $$\left \{ \begin{array}{l} i\partial_tu+\partial_x^2u=\alpha uv,\qquad t\!\in\![-T,T], \ x\!\in\!\mathbb R,\\ \partial_tv+\nu\mathcal H\partial^2_xv=\beta \partial_x(|u|^2),\\ u(0,x)=\phi, \ v(0,x)=\psi, \qquad (\phi,\psi)\!\in\!H^{s}(\mathbb R)\!\times\!H^{s'}\!(\mathbb R). \end{array} \right.$$ In the non-resonant case $(|\nu|\ne1)$, we prove local well-posedness for a large class of initial data. This improves the results obtained by Bekiranov, Ogawa and Ponce (1998). Moreover, we prove $C^2$-ill-posedness at low-regularity, and also when the difference of regularity between the initial data is large enough. As far as we know, this last ill-posedness result is the first of this kind for a nonlinear dispersive system. Finally, we also prove that the local well-posedness result obtained by Pecher (2006) in the resonant case $(|\nu|=1)$ is sharp except for the end-point.