On the dynamics of a quadratic Schr\"odinger system in dimension $n=5$

TL;DR

This paper establishes a precise criterion for the global well-posedness of a quadratic nonlinear Schrödinger system in five dimensions, based on ground state properties and sharp inequalities, advancing understanding of such systems.

Contribution

It provides the first sharp criterion for global well-posedness in energy space for quadratic Schrödinger systems in five dimensions, using ground state analysis and functional inequalities.

Findings

Derived a sharp Gagliardo-Nirenberg inequality.

Characterized ground states via Weinstein-type functional.

Established a criterion linking charge, energy, and well-posedness.

Abstract

In this work we give a sharp criterion for the global well-posedness, in the energy space, for a system of nonlinear Schr\"odinger equations with quadratic interaction in dimension $n=5$. The criterion is given in terms of the charge and energy of the ground states associated with the system, which are obtained by minimizing a Weinstein-type functional. The main result is then obtained in view of a sharp Gagliardo-Nirenberg-type inequality.