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

Abstract

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 H^s(R)xH^k(R) with (s,k)=(0,-3/4+). This result implies in particular the global well-posedness in the energy regularity (s,k)=(1,1). Both results considerably improve the previous ones by Bekiranov, Ogawa and Ponce (1997), Guo and Miao (1999), and Tsutsumi (1993).