Dynamics of focusing nonlinear Schr\"odinger equation with partial harmonic confinement in higher dimensions

TL;DR

This paper extends the understanding of the focusing nonlinear Schrödinger equation with partial harmonic confinement to higher dimensions, overcoming previous limitations by introducing new analytical strategies.

Contribution

The authors develop a novel approach that bypasses the concentration-compactness method, enabling sharp scattering results in higher dimensions for the model.

Findings

Established sharp scattering results in higher dimensions.

Introduced interaction Morawetz-Dodson-Murphy estimates.

Provided an alternative variational characterization of the ground state.

Abstract

We study the following focusing intercritical nonlinear Schr\"odinger equation with partial harmonic confinement: \begin{equation*} \begin{cases} i\partial_t u+\Delta_{z}u-y^2 u =- |u|^{\alpha}u,\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 $\alpha$ satisfies \begin{equation}\label{assumption} \frac{4}{d}< \alpha<\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.