Local well-posedness for the Schr\"{o}dinger-KdV system in $H^{s_1}\times H^{s_2}$

Abstract

In this paper, we study local well-posedness theory of the Cauchy problem for Schr\"{o}dinger-KdV system in Sobolev spaces $H^{s_1}\times H^{s_2}$. We obtain the local well-posedness when $s_1\geq 0$, $\max\{-3/4,s_1-3\}\leq s_2\leq \min\{4s_1,s_1+2\}$. The result is sharp in some sense and improves previous one by Corcho-Linares \cite{corcho2007well}. The endpoint case $(s_1,s_2) = (0,-3/4)$ has been solved in \cite{guo2010well,wang2011cauchy}. We show the necessary and sufficient conditions for related estimates in Bourgain spaces. To solve the borderline cases, we use the $U^p-V^p$ spaces introduced by Koch-Tataru \cite{kochtataru} and function spaces constructed by Guo-Wang \cite{guo2010well}. We also use normal form argument to control the nonresonant interaction.