Schr\"odinger-improved Boussinesq system in two space dimensions

TL;DR

This paper establishes the global existence and uniqueness of strong solutions for a two-dimensional Schr"odinger-improved Boussinesq system and analyzes its limit behavior as the improvement parameter vanishes, connecting it to the Zakharov system.

Contribution

It proves the well-posedness of the Schr"odinger-improved Boussinesq system in 2D without small data restrictions and explores the system's limit to the Zakharov equations.

Findings

Global strong solutions exist and are unique without smallness assumptions.

Solutions converge to the Zakharov system as the improvement parameter tends to zero.

The vanishing limit connects the Boussinesq system to the Zakharov system under certain conditions.

Abstract

We study the Cauchy problem for the Schr\"odinger-improved Boussinesq system in a two dimensional domain. Under natural assumptions on the data without smallness, we prove the existence and uniqueness of global strong solutions. Moreover, we consider the vanishing "improvement" limit of global solutions as the coefficient of the linear term of the highest order in the equation of ion sound waves tends to zero. Under the same smallness assumption on the data as in the Zakharov case, solutions in the vanishing "improvement" limit are shown to satisfy the Zakharov system.