The Cauchy problem for the Schrödinger–KdV system

Abstract

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 L 2 × H − 3 4 , and if β = 0 then it is locally well-posed in H s × H − 3 4 with − 3 16 < s ⩽ 1 4 . 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 G s × F s , where G s and F s are dyadic Bourgain-type spaces related to the Schrödinger operator i ∂ t + ∂ x 2 and the Airy operator ∂ t + ∂ x 3 , respectively, but with a modification on F s in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.