Threshold solutions for the energy-critical NLS system with quadratic interaction

TL;DR

This paper classifies the behavior of radial solutions to a six-dimensional energy-critical quadratic nonlinear Schrödinger system at the threshold energy level, extending previous classifications below this energy.

Contribution

It provides a detailed classification of solution dynamics at the critical energy level for the first time in this context.

Findings

Classified solution behaviors at the threshold energy

Extended understanding of energy-critical NLS systems

Identified conditions for solution blow-up or scattering

Abstract

In this paper, we study the Cauchy problem for a quadratic nonlinear Schr\"{o}dinger system in dimension six. In~\cite{GaoMengXuZheng}, the authors classified the behavior of solutions under the energy constraint $E(u) < E(Q)$, where $Q$ denotes the ground state. In this work, we classify the dynamics of radial solutions at the threshold energy $E(u) = E(Q)$.