On travelling waves of the non linear Schr\"odinger equation escaping a potential well

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.