Local well-posedness for the Schrödinger-KdV system in $H^{s_1}\times H^{s_2}$, II

Abstract

In this paper, we continue the study of the local well-posedness theory for the Schrödinger-KdV system in the Sobolev space $H^{s_1}\times H^{s_2}$. We show the local well-posedness in $H^{-3/16}\times H^{-3/4}$ for $\beta =0$. Combining our work \cite{banchenzhang}, we also have the local well-posedness for $\max \{-3/4,s_1-3\}\le s_2\le \min \{4s_1,s_1+2\}$. The result is sharp by using the contraction mapping argument.