On the energy-critical quadratic nonlinear Schr\"odinger system with three waves

TL;DR

This paper studies the global behavior of a three-wave energy-critical quadratic nonlinear Schrödinger system in six dimensions, proving scattering results using concentration compactness and discovering new conserved quantities.

Contribution

It establishes scattering for the system with mass-resonance or radial data below the ground state, introducing new conserved quantities crucial for the analysis.

Findings

Proves scattering for the system below the ground state.

Identifies new conserved quantities in the system.

Extends understanding of energy-critical Schrödinger systems.

Abstract

In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schr\"odinger system $\[ \left\{ \begin{aligned} & i u^1_t + \kappa_1 \Delta u^1 = -\overline{u^2}u^3, \\ & i u^2_t + \kappa_2 \Delta u^2 = -\overline{u^1}u^3, \\ & i u^3_t + \kappa_3 \Delta u^3 = -u^1u^2, \\ \end{aligned} \right. \qquad (t, x) \in \R \times \R^6 \] in energy-space $ {\dot H}^1 \times {\dot H}^1\times{\dot H}^1 $, where the sign of potential energy can not be determined. We prove the scattering theory with mass-resonance (or with radial initial data) below ground state via concentration compactness method. We discover a family of new physically conserved quantities with mass-resonance which play an important role in the proof of scattering.