Fast soliton scattering by attractive delta impurities

TL;DR

This paper analyzes how high-velocity solitons scatter off attractive delta impurities in the Gross-Pitaevskii equation, providing explicit formulas for reflection and transmission, and showing minimal trapping despite bound states.

Contribution

It offers explicit analytic formulas for soliton scattering by attractive delta potentials and demonstrates the vanishing trapping effect at high velocities.

Findings

Reflected and transmitted soliton components are explicitly characterized.

The trapped portion of the soliton vanishes as velocity increases.

The analysis handles difficulties posed by bound states in the attractive potential.

Abstract

We study the Gross-Pitaevskii equation with an attractive delta function potential and show that in the high velocity limit an incident soliton is split into reflected and transmitted soliton components plus a small amount of dispersion. We give explicit analytic formulas for the reflected and transmitted portions, while the remainder takes the form of an error. Although the existence of a bound state for this potential introduces difficulties not present in the case of a repulsive potential, we show that the proportion of the soliton which is trapped at the origin vanishes in the limit.