First quantum corrections for a hydrodynamics of a nonideal Bose gas

In the paper we consider a hydrodynamical description of a nonideal Bose gas in one-loop approximation. We calculate an effective action which consists of mean field contributions and first quantum correction. This provides the equations of motion for the density and velocity of the gas where both mean field contributions and fluctuations are presented. To fulfill the calculation we make use of the formalism of functional integrals to map the problem to a problem of quantum gravity to benefit from a well-developed technique in this field. This effective action provides all correlation functions for the system and is a basis for a consideration of dynamics of the gas. Response functions are briefly discussed. Applications to the trapped bosons are reviewed. Together with Ref.\cite{IS} the paper provide complete description of the condensate fraction and the deplition in the case of Bose condensed gases.