Sharp local well-posedness for the Schr\"odinger-Korteweg-de Vries system

TL;DR

This paper establishes sharp local well-posedness results for the Schrödinger-Korteweg-de Vries system using integrated-by-parts solutions and frequency estimates, extending global results to lower regularities.

Contribution

It introduces the concept of integrated-by-parts strong solutions and extends global well-posedness to regularities above 1/2.

Findings

Proved sharp local existence in $H^k \times H^s$ spaces.

Extended global well-posedness to $k,s > 1/2$.

Developed frequency-restricted estimates for the system.

Abstract

We prove a sharp local existence result for the Schr\"odinger-Korteweg-de Vries system with initial data in $H^k(\mathbb{R})\times H^s(\mathbb{R})$. The proof is based on the concept of \textit{integrated-by-parts strong solution}, which generalizes the classical notion of strong solution, and on frequency-restricted estimates. Moreover, we extend the known global well-posedness result to regularities $k,s>1/2$.