Uniqueness and orbital stability of standing waves for the nonlinear Schrodinger equation with a partial confinement

TL;DR

This paper studies the uniqueness and stability of ground states for a 3D nonlinear Schrödinger equation with partial confinement, revealing how increasing confinement leads to dimension reduction and ensuring the ground state's uniqueness and orbital stability.

Contribution

It establishes the convergence rate of 3D ground states to 1D states under strong confinement and proves the uniqueness and orbital stability of the 3D minimizer.

Findings

1d ground state derived from 3d minimizer with explicit convergence rate

Uniqueness of 3d minimizer under strong confinement

Orbital stability of the minimizer

Abstract

We consider the 3d cubic nonlinear Schr\"odinger equation (NLS) with a strong 2d harmonic potential. The model is physically relevant to observe the lower-dimensional dynamics of the Bose-Einstein condensate, but its ground state cannot be constructed by the standard method due to its supercritical nature. In Bellazzini-Boussa\"id-Jeanjean-Visciglia \cite{BBJV}, the authors constructed a proper ground state introducing a constrained energy minimization problem. In this paper, we further investigate the properties of the ground state. First, we show that as the partial confinement is increased, the 1d ground state is derived from the 3d energy minimizer with a precise rate of convergence. Then, by employing this dimension reduction limit, we prove the uniqueness of the 3d minimizer provided that the confinement is sufficiently strong. Consequently, we obtain the orbital stability of the minimizer, which improves that of the set of minimizers in the previous work \cite{BBJV}.