  We reformulate the Gross-Pitaevskii equation with an external parabolic
potential as a discrete dynamical system, by using the basis of Hermite
functions. We consider small amplitude stationary solutions with a single node,
called dark solitons, and examine their existence and linear stability.
Furthermore, we prove the persistence of a periodic motion in a neighborhood of
such solutions. Our results are corroborated by numerical computations
elucidating the existence, linear stability and dynamics of the relevant
solutions.
