Dynamics of the Energy-Critical Nonlinear Schr\"{o}dinger System in ${\mathbb R}^{4}$

TL;DR

This paper studies the behavior of radial solutions at threshold energy for a 3-component energy-critical nonlinear Schrödinger system in four dimensions, highlighting new spectral challenges and establishing solution uniqueness.

Contribution

It introduces a novel analysis of the linearized operator with a 2-dimensional kernel and develops a new modulation parameter for the system.

Findings

Detailed coercivity properties of the linearized operators are established.

A new modulation parameter is introduced to handle the additional eigenfunction.

Uniqueness of exponentially decaying solutions to the linearized equation is proven.

Abstract

In this paper, we investigate the dynamics of radial solutions at threshold energy for a 3-component Schr\"{o}dinger system with cubic nonlinearity in four dimensions. The main difference from the cases previously addressed in the literature is that, in our system, the kernel of the imaginary part $L_I$ of the linearized operator $-i{\mathcal L}=L_{R}+iL_{I}$ has dimension 2. To overcome this difficulty, we carry out a detailed study of the coercivity properties of these operators. We also introduce a new modulation parameter associated with the additional eigenfunction in the kernel of the operator $L_{I}$, which enables us to perform the modulation analysis and establish the uniqueness of exponentially decaying solutions to the linearized equation.