Scattering For Mass-resonance Nonlinear Schr\"odinger System in 5D

TL;DR

This paper simplifies the proof of scattering for a 5D mass-resonance nonlinear Schrödinger system by using a new method, establishing a scattering criterion, and excluding mass concentration through decay estimates.

Contribution

It provides a simplified proof of scattering for the system, introducing a new criterion and techniques to exclude mass concentration, improving upon previous methods.

Findings

Established a scattering criterion in H^1(R^5) x H^1(R^5)

Used interaction Morawetz estimate and Galilean transform to exclude mass concentration

Simplified the proof of scattering theory for the system

Abstract

In this paper, we simplify the proof of M. Hamano in \cite{Hamano2018}, scattering theory of the solution to \eqref{NLS system}, by using the method from B. Dodson and J. Murphy in \cite{Dodson2018}. Firstly, we establish a criterion to ensure the solution scatters in $ H^1(\mathbb{R}^5) \times H^1(\mathbb{R}^5) $. In order to verify the correctness of the condition in scattering criterion, we must exclude the concentration of mass near the origin. The interaction Morawetz estimate and Galilean transform characterize a decay estimate, which implies that the mass of the system cannot be concentrated.