The Cauchy problem of the Schrödinger-Korteweg-de Vries system

Abstract

We study the Cauchy problem of the Schrödinger-Korteweg-de Vries system. First, we establish 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. In particular, we obtain local well posedness for the initial data in $H^{-\frac{3}{16}+}({\mathbb{R}})\times H^{-\frac{3}{4}+}({\mathbb{R}})$ in the resonant case; it is almost optimal except for the endpoint. Finally, we establish global well-posedness results in $H^s({\mathbb{R}})\times H^s({\mathbb{R}})$ when $s>\frac{1}{2}$ regardless of whether we are in the resonant case or in the non-resonant case, which improves the results of Pecher (2005).