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 $H^{-{3/16}+}(\R)\times H^{-{3/4}+}(\R)$ in the resonant case, it is almost the optimal except the endpoint. At last we establish the global well-posedness results in $H^s(\R)\times H^s(\R)$ when $s>\dfrac{1}{2}$ no matter in the resonant case or in the non-resonant case, which improve the results of Pecher (2005).