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.

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关于平面中带相反电荷的困涡的一些构型

我们目前工作的目标是确定所有可能的驻波配置，涉及原子玻色 - 爱因斯坦凝聚态 (BEC) 中不同电荷的少数涡流。在这项工作中，我们部署了计算代数方法的使用，以识别具有多达 6 个涡旋的静止多涡旋状态。使用不变量和对称性可以推导出一组基本对称多项式中的方程，然后可以通过 Maple 中的计算代数包完全求解。我们检索了许多先前确定的配置，包括共线配置和多边形（例如四极和六边形）配置。然而，重要的是，我们还检索了具有 4 个正电荷和 2 个负电荷的配置，据我们所知，这在 BEC 研究中是前所未有的。