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Bifurcations of multi-vortex configurations in rotating Bose-Einstein condensates. (English) Zbl 1382.35264

Summary: 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 (\(m_0 \geq 2\)).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B32 Bifurcations in context of PDEs
35R09 Integro-partial differential equations
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
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