Hydrodynamic modes of a one-dimensional trapped Bose gas

We consider two regimes where a trapped Bose gas behaves as a one-dimensional (ID) system. In the first one the Bose gas is microscopically described by 3D mean-field theory, but the trap is so elongated that it behaves as a 1D gas with respect to low-frequency collective modes. In the second regime we assume that the 1D gas is truly 1D and that it is properly described by the Lieb-Liniger model. In both regimes we find the frequency of the lowest compressional mode by solving the hydrodynamic equations. This is done by making use of a method which allows us to find analytical or quasianalytical solutions of these equations for a large class of models approaching very closely the actual equation of state of the Bose gas. We find an excellent agreement with the recent results of Menotti and Stringari obtained from a sum-rule approach.