Self-Bound vortex states in nonlinear Schr\"odinger equations with LHY correction

TL;DR

This paper investigates vortex states in a nonlinear Schrödinger equation with Lee-Huang-Yang correction, establishing existence, stability, and properties of self-bound quantum droplets with vorticity in 2D and 3D.

Contribution

It proves the global existence of solutions and the existence of vortex quantum droplets using constrained energy minimization, linking to inverse-square potential NLS.

Findings

Existence of self-bound vortex quantum droplets in 2D and 3D.

Global well-posedness of the cubic-quartic NLS with LHY correction.

Orbital stability of vortex states under the flow.

Abstract

We study the cubic-quartic nonlinear Schr\"odinger equation (NLS) in two and three spatial dimension. This equation arises in the mean-field description of Bose-Einstein condensates with Lee-Huang-Yang correction. We first prove global existence of solutions in natural energy spaces which allow for the description of self-bound quantum droplets with vorticity. Existence of such droplets, described as central vortex states in 2D and 3D, is then proved using an approach via constrained energy minimizers. A natural connection to the NLS with repulsive inverse-square potential in 2D arises, leading to an orbital stability result under the corresponding flow.