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On the energy-critical quadratic nonlinear Schrödinger system with three waves

Fanfei Meng, Sheng Wang, Chengbin Xu

Abstract

In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schrödinger system $\[ \left\{ \begin{aligned} & i u^1_t + κ_1 Δu^1 = -\overline{u^2}u^3, \\ & i u^2_t + κ_2 Δu^2 = -\overline{u^1}u^3, \\ & i u^3_t + κ_3 Δu^3 = -u^1u^2, \\ \end{aligned} \right. \qquad (t, x) \in \R \times \R^6 \] in energy-space $ {\dot H}^1 \times {\dot H}^1\times{\dot H}^1 $, where the sign of potential energy can not be determined. We prove the scattering theory with mass-resonance (or with radial initial data) below ground state via concentration compactness method. We discover a family of new physically conserved quantities with mass-resonance which play an important role in the proof of scattering.

On the energy-critical quadratic nonlinear Schrödinger system with three waves

Abstract

In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schrödinger system {\dot H}^1 \times {\dot H}^1\times{\dot H}^1 $, where the sign of potential energy can not be determined. We prove the scattering theory with mass-resonance (or with radial initial data) below ground state via concentration compactness method. We discover a family of new physically conserved quantities with mass-resonance which play an important role in the proof of scattering.

Paper Structure

This paper contains 25 sections, 38 theorems, 284 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Methods and results
  4. Outline of the article
  5. Preliminaries
  6. Basic harmonic analysis
  7. Local well-posedness
  8. Profiles decomposition
  9. Virial identity
  10. Variational Analysis
  11. Ground state
  12. Energy trapping
  13. Global Well-Posedness and Scattering
  14. Existence of critical solution
  15. Finite time blow-up
  16. ...and 10 more sections

Key Result

Theorem 1.9

Let $\boldsymbol{\rm u}$ : $I\times \mathbb{R}^6 \to \mathbb{C}^3$ be a solution to NLS system satisfying the mass-resonance (or $\boldsymbol{\rm u}$ is radial without mass-resonance). If then

Theorems & Definitions (72)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4: Scattering size, Killip2010
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7: ${\rm\dot H}^1$ scattering
  • Remark 1.8
  • Theorem 1.9: Bounds of scattering size
  • Corollary 1.10: Scattering
  • ...and 62 more