Dark matter-wave solitons in the dimensionality crossover

We consider the statics and dynamics of dark matter-wave solitons in the dimensionality crossover regime from three dimensions (3D) to one dimension (1D). There, using the nonpolynomial Schr\"odinger mean-field model, we find that the anomalous mode of the Bogoliubov spectrum has an eigenfrequency which coincides with the soliton oscillation frequency obtained by the 3D Gross-Pitaevskii model. We show that substantial deviations (of the order of 10% or more) from the characteristic frequency ωz∕√2 (ωz being the longitudinal trap frequency) are possible even in the purely 1D regime.