We analyze global bifurcations along the family of radially symmetric vortices in the Gross–Pitaevskii equation with a symmetric harmonic potential and a chemical potential µ under the steady rotation with frequency \({\Omega}\). The families are constructed in the small-amplitude limit when the chemical potential µ is close to an eigenvalue of the Schrö dinger operator for a quantum harmonic oscillator. We show that for \({\Omega}\) near 0, the Hessian operator at the radially symmetric vortex of charge \({m_0 \in \mathbb{N}}\) has m ₀(m ₀+1)/2 pairs of negative eigenvalues. When the parameter \({\Omega}\) is increased, 1+m ₀(m ₀-1)/2 global bifurcations happen. Each bifurcation results in the disappearance of a pair of negative eigenvalues in the Hessian operator at the radially symmetric vortex. The distributions of vortices in the bifurcating families are analyzed by using symmetries of the Gross–Pitaevskii equation and the zeros of Hermite–Gauss eigenfunctions. The vortex configurations that can be found in the bifurcating families are the asymmetric vortex (m ₀ = 1), the asymmetric vortex pair (m ₀ = 2), and the vortex polygons \({(m_0 \geq 2)}\).

中文翻译：

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旋转玻色－爱因斯坦凝聚物中的多涡旋构型的分叉

我们分析了在稳定旋转频率为（{\ Omega} \）下，Gross–Pitaevskii方程中具有对称谐波势和化学势µ的径向对称涡旋族的整​​体分叉。当化学势μ接近于量子谐波振荡器的Schrödinger算子的特征值时，这些族在小振幅范围内构建。我们证明，对于\（{\ Omega} \）接近0，在径向对称的电荷\（{m_0 \ in \ mathbb {N}} \）中的Hessian算子具有m ₀（m ₀ +1）/ 2对负特征值。当参数\（{\ Omega} \） 增加1+ m ₀（m ₀ -1）/ 2全局分支。每次分叉都会导致Hessian算子在径向对称涡旋中一对负特征值的消失。利用Gross-Pitaevskii方程的对称性和Hermite-Gauss本征函数的零点，分析了分叉族中的涡旋分布。可以在分叉族中找到的涡旋构型是非对称涡旋（m ₀ = 1），非对称涡旋对（m ₀ = 2）和涡旋多边形\（{（m_0 \ geq 2）} \）。