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

TL;DR

This paper advances the understanding of the local well-posedness of the Schrödinger-KdV system in specific Sobolev spaces, establishing new results for certain regularity levels and confirming their sharpness.

Contribution

It extends previous work by proving local well-posedness in new Sobolev spaces and identifying the sharpness of these results using contraction mapping.

Findings

Local well-posedness in H^{-3/16}×H^{-3/4} for β=0.

Extended well-posedness results for a range of Sobolev spaces.

Results are sharp based on contraction mapping arguments.

Abstract

In this paper, we continue the study of the local well-posedness theory for the Schr\"{o}dinger-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 $\max\{-3/4,s_1-3\}\leq s_2\leq \min\{4s_1,s_1+2\}$. The result is sharp by using the contraction mapping argument.