The Cauchy Problem of the Schr\"odinger-Korteweg-de Vries System

We study the Cauchy problem of the Schr"odinger-Korteweg-de Vries system. First, we establish the local well-posedness results, which improve the results of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results, which show that they are sharp in some well-posedness thresholds. Particularly, we obtain the local well-posedness for the initial data in $H^{-3\mathrm{/}16+}(\mathbb{R})\times H^{-3\mathrm{/}4+}(\mathbb{R})$ in the resonant case, it is almost the optimal except the endpoint. At last we establish the global well-posedness results in $H^s(\mathbb{R})\times H^s(\mathbb{R})$ when $s>{\displaystyle \frac{1}{2}}$ no matter in the resonant case or in the non-resonant case, which improve the results of Pecher (2005).