Sharp local well-posedness for the Hirota-Satsuma system

TL;DR

This paper proves sharp local and global well-posedness results for the Hirota-Satsuma system in various Sobolev spaces, extending previous diagonal case results using advanced Fourier analysis techniques.

Contribution

It generalizes existing well-posedness results to off-diagonal Sobolev spaces and introduces a new framework combining Fourier restriction norms with integrated-by-parts solutions.

Findings

Established sharp local existence in H^k x H^s spaces depending on dispersion ratio.

Extended global well-posedness theory to off-diagonal Sobolev regimes.

Utilized Fourier restriction norm method with integrated-by-parts strong solutions.

Abstract

We establish sharp local existence results for the Hirota-Satsuma system in $H^k(\mathbb{R}) \times H^s(\mathbb{R})$, depending on the ratio between the dispersion of the components. These theorems significantly generalize previous works, which were restricted to the diagonal case of equal regularity $s=k$. Furthermore, we extend the known global well-posedness theory to the off-diagonal regime. The argument relies on the Fourier restriction norm method coupled with the concept of integrated-by-parts strong solution - a framework that generalizes the classical notion of strong solution.