On Some Configurations of Oppositely Charged Trapped Vortices in the Plane

TL;DR

This paper systematically classifies and discovers new vortex configurations in Bose-Einstein condensates using algebraic and numerical methods, revealing previously unknown multi-vortex arrangements with mixed charges.

Contribution

It introduces a computational algebra approach to identify all stationary multi-vortex states up to six vortices, including a novel configuration with four positive and two negative charges.

Findings

Recovered known vortex configurations such as collinear and polygonal arrangements.

Discovered a new vortex configuration with 4 positive and 2 negative charges.

Validated predictions through numerical simulations of the Gross-Pitaevskii equation.

Abstract

Our aim in the present work is to identify all the possible standing wave configurations involving few vortices of different charges in an atomic Bose-Einstein condensate (BEC). In this effort, we deploy the use of a computational algebra approach in order to identify stationary multi-vortex states with up to 6 vortices. The use of invariants and symmetries enables deducing a set of equations in elementary symmetric polynomials, which can then be fully solved via computational algebra packages within Maple. We retrieve a number of previously identified configurations, including collinear ones and polygonal (e.g. quadrupolar and hexagonal) ones. However, importantly, we also retrieve a configuration with 4 positive charges and 2 negative ones which is unprecedented, to the best of our knowledge, in BEC studies. We corroborate these predictions via numerical computations in the fully two-dimensional PDE system of the Gross-Pitaevskii type which characterizes the BEC at the mean-field level.