Bifurcations of multi-vortex configurations in rotating Bose--Einstein condensates

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 $\mu$ under the steady rotation with frequency $\Omega$. The families are constructed in the small-amplitude limit when the chemical potential $\mu$ is close to an eigenvalue of the Schr\"{o}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 $(m_0 \geq 2)$.