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 $\mu$ 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 Schrödinger 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 $(m_0\ge 2)$.

References in corpus (4)

• Guiding-center dynamics of vortex dipoles in Bose-Einstein condensates
• Size and dynamics of vortex dipoles in dilute Bose-Einstein condensates
• On the Cubic Lowest Landau Level Equation
• On the characterization of vortex configurations in the steady rotating Bose--Einstein condensates

Cited by in corpus (4)

• Stability and instability properties of rotating Bose-Einstein condensates
• Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with Deflated continuation
• On the Cubic Lowest Landau Level Equation
• On the characterization of vortex configurations in the steady rotating Bose--Einstein condensates