  In the first part of this contribution we prove the global existence and
uniqueness of a trajectory that globally converges to the minimizer of the
Gross-Pitaevskii energy functional for a large class of external potentials.
Using the method of Sobolev gradients we can provide an explicit construction
of this minimizing sequence.
  In the second part we numerically apply these results to a specific
realization of the external potential and illustrate the main benefits of the
method of Sobolev gradients, which are high numerical stability and rapid
convergence towards the minimizer.
