  A unified model of a dilute Bose-Einstein condensate is proposed, combining
of the logarithmic and Gross-Pitaevskii nonlinear terms in a wave equation,
where the Gross-Pitaevskii term describes two-body interactions, as suggested
by standard perturbation theory; while the logarithmic term is essentially
non-perturbative, and takes into account quantum vacuum effects. The model is
shown to have excellent agreement with sound propagation data in the condensate
of cold sodium atoms known since the now classic works by Andrews and
collaborators. The data also allowed us to place constraints on two of the
unified model's parameters, which describe the strengths of the logarithmic and
Gross-Pitaevskii terms. Additionally, we suggest an experiment constraining the
value of the third parameter (the characteristic density scale of the
logarithmic part of the model), using the conjectured attraction-repulsion
transition of many-body interaction inside the condensate.
