Scattering For three waves Nonlinear Schr\"odinger System with mass-resonance in 5D

TL;DR

This paper investigates the long-term scattering behavior of a three-wave nonlinear Schrödinger system in five dimensions, establishing new estimates and criteria for scattering in the energy space, with results depending on radial symmetry and mass-resonance conditions.

Contribution

It introduces an interaction Morawetz estimate for the three-wave NLS system and extends scattering results under certain symmetry and resonance assumptions.

Findings

Established scattering criteria in energy space with mass-resonance.

Removed mass-resonance condition under radial symmetry.

Developed an interaction Morawetz estimate for the system.

Abstract

In this paper, we study the dynamics behavior of the NLS system with three waves interaction in the energy space $H^1(\mathbb{R}^5) \times H^1(\mathbb{R}^5)\times H^1(\mathbb{R}^5) $. Inspired by B. Dodson and J. Murphy in \cite{Dodson2018}, we establish an interaction Morawetz estimate for the NLS system, together with the criterion which proved by Tao-Dodson--Murphy we can get the scattering under the ground state in energy space with mass-resonance. Under the radial assumption, we can remove the mass-resonance condition.