Bifurcations of Multi-Vortex Configurations in Rotating Bose-Einstein Condensates

摘要

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 A mu under the steady rotation with frequency . The families are constructed in the small-amplitude limit when the chemical potential A mu is close to an eigenvalue of the Schro dinger operator for a quantum harmonic oscillator. We show that for near 0, the Hessian operator at the radially symmetric vortex of charge has m (0)(m (0)+1)/2 pairs of negative eigenvalues. When the parameter 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 .