Crossover dark soliton dynamics in ultracold one-dimensional Bose gases

Ultracold confined one-dimensional atomic gases are predicted to support dark soliton solutions arising from a nonlinear Schr\"{o}dinger equation of suitable nonlinearity. In weakly-interacting (high density) gases, the nonlinearity is cubic, whereas an approximate model for describing the behaviour of strongly - interacting (low density) gases is one characterized by a quintic nonlinearity. We use an approximate analytical expression for the form of the nonlinearity in the intermediate regimes to show that, near the crossover between the two different regimes, the soliton is predicted and numerically confirmed to oscillate at a frequency of $\sqrt{2/3}\Omega$, where $\Omega$ is the harmonic trap frequency.