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Dynamics of focusing nonlinear Schrödinger equation with partial harmonic confinement in higher dimensions

Tianhao Liu, Zuyu Ma, Yilin Song, Jiqiang Zheng

Abstract

We study the following focusing intercritical nonlinear Schrödinger equation with partial harmonic confinement: \begin{equation*} \begin{cases} i\partial_t u+Δ_{z}u-y^2 u =- |u|^αu,\quad t\in \mathbb{R},\newline u(0,z)= u_0(z), \ z=(x,y)\in \mathbb{R}^d\times \mathbb{R}, \end{cases} \end{equation*} where $d \geq 1 $ is an integer and the exponent $α$ satisfies \begin{equation}\label{assumption} \frac{4}{d}< α<\begin{cases} \frac{4}{d-1}, \,\,\, \text{if} ~~ d\geq 2; \newline + \infty,\,\,\, \text{if} ~~ d=1. \end{cases} \end{equation} For this model, A. Ardia and R. Carles [Comm. Math. Sci. 19 (2021), 993-1032] established a sharp scattering result below the ground state threshold in dimensions $d \leq 4$ via the concentration-compactness and rigidity argument. However, their approach breaks down in higher dimensions due to the lack of smoothness in the nonlinearity. In this paper, we introduce a new strategy that removes this dimensional restriction and extend their results to higher dimensions by circumventing the concentration-compactness principle. The main ingredients of our work are the interaction Morawetz-Dodson-Murphy estimates and an alternative variational characterization of the ground state threshold.

Dynamics of focusing nonlinear Schrödinger equation with partial harmonic confinement in higher dimensions

Abstract

We study the following focusing intercritical nonlinear Schrödinger equation with partial harmonic confinement: \begin{equation*} \begin{cases} i\partial_t u+Δ_{z}u-y^2 u =- |u|^αu,\quad t\in \mathbb{R},\newline u(0,z)= u_0(z), \ z=(x,y)\in \mathbb{R}^d\times \mathbb{R}, \end{cases} \end{equation*} where is an integer and the exponent satisfies \begin{equation}\label{assumption} \frac{4}{d}< α<\begin{cases} \frac{4}{d-1}, \,\,\, \text{if} ~~ d\geq 2; \newline + \infty,\,\,\, \text{if} ~~ d=1. \end{cases} \end{equation} For this model, A. Ardia and R. Carles [Comm. Math. Sci. 19 (2021), 993-1032] established a sharp scattering result below the ground state threshold in dimensions via the concentration-compactness and rigidity argument. However, their approach breaks down in higher dimensions due to the lack of smoothness in the nonlinearity. In this paper, we introduce a new strategy that removes this dimensional restriction and extend their results to higher dimensions by circumventing the concentration-compactness principle. The main ingredients of our work are the interaction Morawetz-Dodson-Murphy estimates and an alternative variational characterization of the ground state threshold.

Paper Structure

This paper contains 15 sections, 24 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Structure of the paper
  5. Preliminaries
  6. Fourier transform and functional spaces
  7. Harmonic oscillator and associated functional space
  8. Local theory
  9. Scattering criterion
  10. Variational analysis
  11. preliminary results
  12. The properties of $m_\omega$
  13. The proof of Theorem \ref{['thm groundstate']}
  14. Interaction Morawetz estimate
  15. Proof of the main results

Key Result

Theorem 1.1

Assume that $d\geq 3$ and that assumption holds. Let $u(t)\in C(I,\Sigma )$ be a solution of PHNLS with initial data $u_0=u(0)$, where $I=(-T_-,T_+)$ denotes the maximal lifespan with $T_\pm>0$. Then we have the following dichotomy classification of dynamics.

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1: Moser type estimate, Jao-CPDE
  • Remark 2.1
  • Definition 2.1: Schrödinger admissible pairs
  • Proposition 2.1: Strichartz estimates, Antonelli
  • Lemma 2.2: Exotic Strichartz estimates,TV1TV2
  • Remark 2.2
  • Lemma 2.3
  • ...and 40 more