  We consider the possibility that the dark matter, which is required to
explain the dynamics of the neutral hydrogen clouds at large distances from the
galactic center, could be in the form of a Bose-Einstein condensate. To study
the condensate we use the non-relativistic Gross-Pitaevskii equation. By
introducing the Madelung representation of the wave function, we formulate the
dynamics of the system in terms of the continuity equation and of the
hydrodynamic Euler equations. Hence dark matter can be described as a
non-relativistic, Newtonian Bose-Einstein gravitational condensate gas, whose
density and pressure are related by a barotropic equation of state. In the case
of a condensate with quartic non-linearity, the equation of state is polytropic
with index $n=1$. To test the validity of the model we fit the Newtonian
tangential velocity equation of the model with a sample of rotation curves of
low surface brightness and dwarf galaxies, respectively. We find a very good
agreement between the theoretical rotation curves and the observational data
for the low surface brightness galaxies. The deflection of photons passing
through the dark matter halos is also analyzed, and the bending angle of light
is computed. The bending angle obtained for the Bose-Einstein condensate is
larger than that predicted by standard general relativistic and dark matter
models. Therefore the study of the light deflection by galaxies and the
gravitational lensing could discriminate between the Bose-Einstein condensate
dark matter model and other dark matter models.
