  In this paper we consider the NLS equation with focusing nonlinearities in
the presence of a potential. We investigate the compact soliton motions that
correspond to a free soliton escaping the well created by the potential. We
exhibit the dynamical system driving the exiting trajectory and construct
associated nonlinear dynamics for untrapped motions. We show that the nature of
the potential/soliton is fundamental, and two regimes may exist: one where the
tail of the potential is fat and dictates the motion, one where the tail is
weak and the soliton self interacts with the potential defects, hence leading
to different motions.
