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 in their 2007 paper “Fast soliton scattering by delta impurities,” and the work of Datchev and Holmer, who considered the case of the delta potential with a linear bound state in their 2009 paper “Fast soliton scattering by attractive delta impurities.”

This is joint work with Jason Murphy.