  Bose-Einstein condensates with an attractive 1/r interaction and with
dipole-dipole interaction are investigated in the framework of the Gaussian
variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A,
submitted]. We demonstrate that the method of coupled Gaussian wave packets is
a full-fledged alternative to direct numerical solutions of the
Gross-Pitaevskii equation, or even superior in that coupled Gaussians are
capable of producing both, stable and unstable states of the Gross-Pitaevskii
equation, and thus of giving access to yet unexplored regions of the space of
solutions of the Gross-Pitaevskii equation. As an alternative to numerical
solutions of the Bogoliubov-de Gennes equations, the stability of the
stationary condensate wave functions is investigated by analyzing the stability
properties of the dynamical equations of motion for the Gaussian variational
parameters in the local vicinity of the stationary fixed points. For
blood-cell-shaped dipolar condensates it is shown that on the route to collapse
the condensate passes through a pitchfork bifurcation, where the ground state
itself turns unstable, before it finally vanishes in a tangent bifurcation.
