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

Abstract

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 Schro 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 $${(m_0 \geq 2)}$$ .