We consider the dynamics of a boosted soliton evolving under the cubic NLS
with an external potential. We show that for sufficiently large velocities, the
soliton is effectively transmitted through the potential. This result extends
work of Holmer, Marzuola, and Zworski, who considered the case of a delta
potential with no bound states, and work of Datchev and Holmer, who considered
the case of a delta potential with a linear bound state.Comment: 13 page