On the well-posedness of the Schrödinger–Korteweg–de Vries system

Abstract

We prove that the Cauchy problem for the Schrödinger–Korteweg–de Vries system is locally well-posed for the initial data belonging to the Sobolev spaces L 2 ( R ) × H − 3 / 4 ( R ) , and H s ( R ) × H − 3 / 4 ( R ) ( s > − 1 / 16 ) for the resonant case. The new ingredient is that we use the F ¯ s -type space, introduced by the first author in Guo (2009) [10], to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares (2007) [6].