  The Gross-Pitaevskii (GP) equation is a long-wavelength approach widely used
to describe the dilute Bose-Einstein condensates (BEC). However, in many
physical situations, such as higher densities, this approximation unlikely
suffices hence one might need models which would account for long-range
correlations and multi-body interactions. We show that the Bose liquid
described by the logarithmic wave equation has a number of drastic differences
from the GP one. It possesses the self-sustainability property: while the free
GP condensate tends to spill all over the available volume the logarithmic one
tends to form a Gaussian-type droplet - even in the absence of an external
trapping potential. The quasi-particle modes of the logarithmic BEC are shown
to acquire a finite size despite the bare particles being assumed point-like,
i.e., the spatial extent emerges here as a result of quantum many-body
correlations. Finally, we study the elementary excitations and demonstrate that
the background density changes the topological structure of their momentum
space which, in turn, affects their dispersion relations. Depending on the
density the latter can be of the massive relativistic, massless relativistic,
tachyonic and quaternionic type.
