On the Well-posedness of the Schr\"odinger-Korteweg-de Vries system

TL;DR

This paper establishes local well-posedness for the Schr"odinger-Korteweg-de Vries system with initial data in specific Sobolev spaces, utilizing a novel function space approach to handle coupling and scaling challenges.

Contribution

It introduces the use of the $ar{F}^s$ space for the KdV component and provides uniform estimates to address the lack of scaling invariance, improving previous results.

Findings

Proves local well-posedness in Sobolev spaces $L^2 \times H^{-{3/4}}$.

Employs the $ar{F}^s$ space to manage coupling terms.

Establishes uniform estimates for multipliers to overcome scaling issues.

Abstract

We prove that the Cauchy problem for the Schr\"odinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(\R)\times H^{-{3/4}}(\R)$. The new ingredient is that we use the $\bar{F}^s$ type space, introduced by the first author in \cite{G}, 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.