A recent analysis has revealed singular but physically relevant localized two-dimensional vortex states with density ∼r ^−4/3 at and a convergent total norm, which are maintained by the interplay of the potential of the attraction to the center, ∼−r ⁻², and a self-repulsive quartic nonlinearity, produced by the Lee-Huang-Yang correction to the mean-field dynamics of Bose–Einstein condensates. In optics, a similar setting, with the density singularity ∼r ⁻¹, is realized with the help of quintic self-defocusing. Here we present physically relevant antidark singular-vortex states in these systems, existing on top of a flat background. Numerical solutions for them are very accurately approximated by the Thomas–Fermi wave function. Their stability exactly obeys an analytical criterion derived from the consideration of small perturbations. The singular-vortex states exist as well in the case when the effective potential is weakly repulsive. It is demonstrated that the singular vortices can be excited by the input in the form of the ordinary nonsingular vortices, hence the singular modes can be created in the experiment. We also consider regular (dark) vortices maintained by the flat background, under the action of the repulsive central potential ∼+r ⁻². The dark modes with vorticities l = 0 and 1 are completely stable. In the case when the central potential is attractive, but the effective one, which includes the centrifugal term, is repulsive, and, in addition, a weak trapping potential ∼r ² is applied, dark vortices with l = 1 feature an intricate pattern of alternating stability and instability regions. Under the action of the instability, states with l = 1 travel along tangled trajectories, which stay in a finite area defined by the trap. The analysis is also reported for dark vortices with l = 2, which feature a complex structure of alternating intervals of stability and instability against splitting. Lastly, simple but novel flat vortices are found at the border between the anidark and dark ones.

最近的一项分析揭示了奇异但物理相关的局部二维涡旋状态，其密度为 ～r ^-4/3 at和收敛的总范数，这些状态由中心吸引力的相互作用维持，～^- r ^{- 2}，以及自斥四次非线性，由李-黄-杨校正对玻色-爱因斯坦凝聚的平均场动力学产生。在光学中，在五次自散焦的帮助下实现了类似的设置，具有密度奇点∼ r ^-1。在这里，我们展示了与物理相关的反黑暗这些系统中的奇异涡旋状态，存在于平坦的背景之上。它们的数值解由 Thomas-Fermi 波函数非常精确地逼近。它们的稳定性完全符合从考虑小扰动得出的分析标准。在有效势弱排斥的情况下也存在奇异涡旋状态。证明了奇异涡可以被普通非奇异涡形式的输入激发，因此可以在实验中创建奇异模式。我们还考虑在排斥中心势∼+ r ^-2的作用下由平坦背景维持的规则（暗）涡流。具有涡量l的暗模式 = 0 和 1 是完全稳定的。在中心势是有吸引力的情况下，包括离心项在内的有效势是排斥的，此外，在施加弱捕获势 ~ r ²的情况下，l  = 1 的暗涡具有复杂的模式交替稳定和不稳定区域。在不稳定的作用下，l  = 1 的状态沿着纠缠的轨迹行进，这些轨迹停留在陷阱定义的有限区域内。还报告了l  = 2 的暗涡旋的分析，其特征是稳定和不稳定性交替间隔的复杂结构，以防止分裂。最后，简单但新颖的扁平涡流 被发现在anidark和黑暗之间的边界。