The Cauchy problem for the Schrödinger–KdV system

In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger–Korteweg–de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.