Dynamics of the energy-critical nonlinear Schrödinger system in $  \mathbb R^{4} $

Alex H. Ardila

doi:10.3934/dcds.2025180

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2026, Volume 50: 1-45. Doi: 10.3934/dcds.2025180

This volume Previous Article A simple description of blow-up solutions through dynamics at infinity in nonautonomous ODEs Next Article On second-order $ L^\infty $ variational problems with lower-order terms

Dynamics of the energy-critical nonlinear Schrödinger system in $ \mathbb R^{4} $

Alex H. Ardila

Department of Mathematics, Universidad del Valle, Cali, Colombia

Received: May 2025

Revised: September 2025

Early access: November 11, 2025

Published online: November 11, 2025

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Abstract

In this paper, we investigate the dynamics of radial solutions at threshold energy for a 3-component Schrödinger 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.

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Mathematics Subject Classification: Primary: 35Q55.

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