Oscillations of dark solitons in trapped Bose-Einstein condensates

TL;DR

This paper analyzes the oscillatory behavior of dark solitons in trapped Bose-Einstein condensates using asymptotic methods, revealing uniform oscillation frequencies and amplitude decay due to sound emission.

Contribution

It introduces a systematic multi-scale asymptotic expansion approach to predict dark soliton oscillations and their decay in a parabolic trap, extending previous analytical and numerical studies.

Findings

Dark solitons oscillate with a uniform frequency near the trap center.

Amplitude of oscillations decays over time due to radiative losses.

Theoretical predictions agree with numerical simulations.

Abstract

We consider a one-dimensional defocusing Gross--Pitaevskii equation with a parabolic potential. Dark solitons oscillate near the center of the potential trap and their amplitude decays due to radiative losses (sound emission). We develop a systematic asymptotic multi-scale expansion method in the limit when the potential trap is flat. The first-order approximation predicts a uniform frequency of oscillations for the dark soliton of arbitrary amplitude. The second-order approximation predicts the nonlinear growth rate of the oscillation amplitude, which results in decay of the dark soliton. The results are compared with the previous publications and numerical computations.