Transmission of fast solitons for the NLS with an external potential

TL;DR

This paper studies how fast-moving solitons pass through external potentials in the cubic nonlinear Schrödinger equation, showing effective transmission at high velocities, extending previous delta potential results.

Contribution

It extends existing results by analyzing soliton transmission through general external potentials at high velocities, beyond delta potentials.

Findings

Fast solitons are effectively transmitted through external potentials.

Transmission effectiveness increases with soliton velocity.

Results generalize previous delta potential studies.

Abstract

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.