Variational approximations of trapped vortices in the large-density limit

The Gross–Pitaevskii equation with a harmonic potential and repulsive nonlinear interactions is considered in the large-density limit, also known as the Thomas–Fermi limit. In two space dimensions, we employ the Rayleigh–Ritz method to obtain variational approximations of single vortices, dipole pairs and quadrupoles trapped in the harmonic potential. In particular, we compute the eigenfrequency of the single vortex precession about the centre of symmetry of the harmonic potential, as well as the eigenfrequencies of the oscillations of the dipole and quadrupole vortex configurations. The asymptotic results are compared to numerical computations.