Dynamics of shallow dark solitons in a trapped gas of impenetrable bosons

TL;DR

This paper investigates the behavior of shallow dark solitons in a one-dimensional Bose gas in the Tonks-Girardeau regime, using a mean field approach and perturbation methods to derive analytical and numerical results.

Contribution

It introduces a mean field theory with quintic nonlinearity to describe soliton dynamics in a trapped TG gas, deriving a variable-coefficient KdV equation and analytical expressions for soliton oscillations.

Findings

Soliton oscillation frequency matches the trap frequency.

Analytical expressions for soliton and sound evolution are provided.

Numerical predictions agree with analytical results.

Abstract

The dynamics of linear and nonlinear excitations in a Bose gas in the Tonks-Girardeau (TG) regime with longitudinal confinement are studied within a mean field theory of quintic nonlinearity. A reductive perturbation method is used to demonstrate that the dynamics of shallow dark solitons, in the presence of an external potential, can effectively be described by a variable-coefficient Korteweg-de Vries equation. The soliton oscillation frequency is analytically obtained to be equal to the axial trap frequency, in agreement with numerical predictions obtained by Busch {\it et al.} [J. Phys. B {\bf 36}, 2553 (2003)] via the Bose-Fermi mapping. We obtain analytical expressions for the evolution of both soliton and emitted radiation (sound) profiles.