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

Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. II. Applications