Back-reaction of perturbation wave packets on gray solitons

TL;DR

This paper investigates how wave packet perturbations influence gray solitons in the nonlinear Schrödinger equation, revealing a mutual forward shift due to back-reaction effects confirmed by numerical simulations and discussing implications for quantum gases.

Contribution

It provides an exact analysis of wave packet back-reaction on gray solitons and introduces a simple theory to quantify the soliton's forward shift, supported by numerical validation.

Findings

Wave packets effectively jump ahead when passing through a soliton.

The soliton itself experiences a forward shift due to back-reaction.

Numerical simulations confirm the theoretical predictions.

Abstract

Within the Bogoliubov-de Gennes linearization theory of quantum or classical perturbations around a background solution to the one-dimensional nonlinear Schr\"odinger equation, we study the back-reaction of wave packet perturbations on a gray soliton background. From our recently published exact solutions, we determine that a wave packet effectively jumps ahead as it passes through a soliton, emerging with a wavelength-dependent forward translation in comparison to its motion in absence of the soliton. From this and from the full theory's exact momentum conservation, we deduce that post-Bogoliubov back-reaction must include a commensurate forward advance by the soliton itself. We quantify this effect with a simple theory, and confirm that it agrees with full numerical solution of the classical nonlinear Schr\"odinger equation. We briefly discuss the implications of this effect for quantum behavior of solitons in quasi-condensed dilute gases at finite temperature.