Well-posedness for the Schrödinger-Korteweg-de Vries system

We study well-posedness of the Cauchy problem associated to the Schrödinger-Korteweg-de Vries system. We obtain local well-posedness for weak initial data, where the best result obtained is for data in the Sobolev space $L^2({\mathbb{R}})\times H^{-\tfrac{3}{4}+}$. This result implies in particular the global well-posedness in the energy space $H^1({\mathbb{R}})\times H^1({\mathbb{R}})$. Both results considerably improve the previous ones by Bekiranov, Ogawa and Ponce (1997), Guo and Miao (1999), and Tsutsumi (1993).