D.E. Pelinovsky and P.G. Kevrekidis
Variational approximations of trapped vortices
in the large-density limit
Nonlinearity 24 (2011), 1271-1289
Abstract:
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 the
space of two dimensions, we employ the Rayleigh–Ritz approximation 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 center 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 confirmed
by the numerical computations of the vortex states and the linearization thereof.
Keywords:
Gross-Pitaevskii equation, harmonic potential, Thomas-Fermi limit, semi-classical limit,
variational Rayleigh-Ritz method, vortex configurations