Jeans instability and turbulent gravitational collapse of Bose-Einstein
  Condensate dark matter halos
  We consider the Jeans instability and the gravitational collapse of the
rotating Bose-Einstein Condensate dark matter halos, described by the zero
temperature non-relativistic Gross-Pitaevskii equation, with repulsive
inter-particle interactions. In the Madelung representation of the wave
function, the dynamical evolution of the galactic halos is described by the
continuity and the hydrodynamic Euler equations, with the condensed dark matter
satisfying a polytropic equation of state with index $n=1$. By considering
small perturbations of the quantum hydrodynamical equations we obtain the
dispersion relation and the Jeans wave number, which includes the effects of
the vortices (turbulence), of the quantum pressure and of the quantum
potential, respectively. The critical scales above which condensate dark matter
collapses (the Jeans radius and mass) are discussed in detail. We also
investigate the collapse/expansion of rotating condensed dark matter halos, and
we find a family of exact semi-analytical solutions of the hydrodynamic
evolution equations, derived by using the method of separation of variables. An
approximate first order solution of the fluid flow equations is also obtained.
The radial coordinate dependent mass, density and velocity profiles of the
collapsing/expanding condensate dark matter halos are obtained by using
numerical methods.
