Scattering theory For Quadratic Nonlinear Schr\"odinger System in dimension six

TL;DR

This paper classifies the long-term behavior of solutions to a six-dimensional energy-critical quadratic nonlinear Schrödinger system, distinguishing between scattering and blow-up based on initial energy and kinetic energy relative to the ground state.

Contribution

It provides a complete classification of scattering versus blow-up for the energy-critical quadratic NLS system in six dimensions under certain initial conditions.

Findings

Solutions with initial energy below the ground state either scatter or blow up depending on kinetic energy.

The classification applies to radial data and non-radial data satisfying the mass-resonance condition.

The results are obtained using the compactness/rigidity method.

Abstract

In this paper, we study the solutions to the energy-critical quadratic nonlinear Schr\"odinger system in ${\dot H}^1\times{\dot H}^1$, where the sign of its potential energy can not be determined directly. If the initial data ${\rm u}_0$ is radial or non-radial but satisfies the mass-resonance condition, and its energy is below that of the ground state, using the compactness/rigidity method, we give a complete classification of scattering versus blowing-up dichotomies depending on whether the kinetic energy of ${\rm u}_0$ is below or above that of the ground state.