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 $$\{\begin{array}{l}i{\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x^{2}u=\alpha uv,t\text{\hspace{0.17em}⁣}\in \text{\hspace{0.17em}⁣}[-T,T],x\text{\hspace{0.17em}⁣}\in \text{\hspace{0.17em}⁣}\mathbb{R},\\ {\mathrm{\partial }}_{t}v+\nu \mathcal{H}{\mathrm{\partial }}_x^{2}v=\beta {\mathrm{\partial }}_x(\mathrm{\mid }u{\mathrm{\mid }}^2),\\ u(0,x)=\varphi ,v(0,x)=\psi ,(\varphi ,\psi )\text{\hspace{0.17em}⁣}\in \text{\hspace{0.17em}⁣}{H}^s(\mathbb{R})\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}{H}^{s^{\mathrm{\prime }}}\text{\hspace{0.17em}⁣}(\mathbb{R})\mathrm{.}\end{array}$$ In the non-resonant case $(\mathrm{\mid }\nu \mathrm{\mid }\mathrm{\ne }1)$, 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 $(\mathrm{\mid }\nu \mathrm{\mid }=1)$ is sharp except for the end-point.