Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Threshold solutions for the energy-critical NLS system with quadratic interaction

Alex H. Ardila, Liliana Cely, Fanfei Meng

TL;DR

This work analyzes the Cauchy problem for a quadratic energy-critical NLS system in $\mathbb{R}^6$ at the ground-state energy $E(\mathcal{Q})$, classifying all radial threshold dynamics. The authors combine spectral analysis of the linearized operator around the ground state, modulation theory, and virial identity arguments to establish convergence, scattering, or blow-up behavior, and to construct special threshold solutions via a transformation to a related NNLS system. A key novelty is handling the asymmetric quadratic nonlinearity through the $T$-transform, enabling the existence and uniqueness of threshold solutions $W^a$ and the corresponding radial threshold states $\mathcal{G}^\pm$. Consequently, any radial solution with $E(\mathbf{u})=E(\mathcal{Q})$ is shown to follow one of six dynamical paths, with precise compatibility to the ground state orbit and exponential convergence rates where applicable.

Abstract

In this paper, we study the Cauchy problem for a quadratic nonlinear Schrödinger system in dimension six. In~\cite{GaoMengXuZheng}, the authors classified the behavior of solutions under the energy constraint $E(u) < E(Q)$, where $Q$ denotes the ground state. In this work, we classify the dynamics of radial solutions at the threshold energy $E(u) = E(Q)$.

Threshold solutions for the energy-critical NLS system with quadratic interaction

TL;DR

This work analyzes the Cauchy problem for a quadratic energy-critical NLS system in at the ground-state energy , classifying all radial threshold dynamics. The authors combine spectral analysis of the linearized operator around the ground state, modulation theory, and virial identity arguments to establish convergence, scattering, or blow-up behavior, and to construct special threshold solutions via a transformation to a related NNLS system. A key novelty is handling the asymmetric quadratic nonlinearity through the -transform, enabling the existence and uniqueness of threshold solutions and the corresponding radial threshold states . Consequently, any radial solution with is shown to follow one of six dynamical paths, with precise compatibility to the ground state orbit and exponential convergence rates where applicable.

Abstract

In this paper, we study the Cauchy problem for a quadratic nonlinear Schrödinger system in dimension six. In~\cite{GaoMengXuZheng}, the authors classified the behavior of solutions under the energy constraint , where denotes the ground state. In this work, we classify the dynamics of radial solutions at the threshold energy .
Paper Structure (13 sections, 50 theorems, 334 equations)

This paper contains 13 sections, 50 theorems, 334 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Virial identities
  4. Spectral properties of the linearized operator
  5. Modulation analysis
  6. Convergence for subcritical threshold solution
  7. Convergence for supercritical threshold solution
  8. Existence of special solutions
  9. Construction of special solutions
  10. A Uniqueness Result
  11. Proof of the main result
  12. Coercivity
  13. Spectrum of the linearized operator

Key Result

Theorem 1.1

Fix $\kappa>0$. Let $\textbf{u}(t)$ be the corresponding solution to SNLS with radial initial data $\textbf{u}_{0}\in \dot{H}^{1}(\mathbb R^{6})\times \dot{H}^{1}(\mathbb R^{6})$.

Theorems & Definitions (95)

  • Theorem 1.1: Sub-threshold scattering/blows up, GaoMengXuZhengAdemiNoguera2022
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 2.1: see Theorem 2.5 in GaoMengXuZheng
  • Proposition 2.2: Sufficient condition for scattering
  • Lemma 2.3: Long time perturbation theory, see Proposition 2.7 in GaoMengXuZheng
  • Proposition 2.4: see NayaOzaTana
  • Lemma 2.5
  • proof
  • ...and 85 more