Collective oscillations of a one-dimensional trapped Bose-Einstein gas

Starting from the hydrodynamic equations of superfluids, we calculate the frequencies of the collective oscillations of a harmonically trapped Bose-Einstein gas for various one-dimensional configurations at zero temperature. These include the mean-field regime described by Gross-Pitaevskii theory and the beyond-mean-field regime at small densities described by the Lieb-Liniger theory. The relevant combinations of the physical parameters governing the transition between the different regimes as well as the conditions of applicability of the hydrodynamic equations are discussed.