Well-posedness and stability in the periodic case for the Benney system

TL;DR

This paper proves local and global well-posedness for the Benney system in periodic Sobolev spaces, constructs explicit traveling wave solutions, and demonstrates their orbital stability under certain conditions.

Contribution

It establishes the lowest regularity for well-posedness of the Benney system and proves orbital stability of explicit traveling wave solutions.

Findings

Well-posedness in $H^{1/2}\times L^2$ and $H^{1}\times L^2$ spaces.

Existence of explicit periodic traveling waves of dnoidal type.

Orbital stability of these traveling waves under certain conditions.

Abstract

We establish local well-posedness results in weak periodic function spaces for the Cauchy problem of the Benney system. The Sobolev space $H^{1/2}\times L^2$ is the lowest regularity attained and also we cover the energy space $H^{1}\times L^2$, where global well-posedness follows from the conservation laws of the system. Moreover, we show the existence of smooth explicit family of periodic travelling waves of \emph{dnoidal} type and we prove, under certain conditions, that this family is orbitally stable in the energy space.