Gravitational collapse of Bose-Einstein Condensate dark matter halos with logarithmic nonlinearity

If dark matter is composed of massive bosons, a Bose-Einstein Condensation process must have occurred during the cosmological evolution. Therefore, galactic dark matter may be in a form of a condensate, characterized by a strong self-interaction. We consider the possibility that the self-interacting potential of the condensate dark matter is of the logarithmic form. In order to describe the condensate dark matter we use the Gross-Pitaevskii equation with a logarithmic nonlinearity, and the Thomas-Fermi approximation. With the use of the hydrodynamic representation of the Gross-Pitaevskii equation we obtain the equation of state of the condensate, which has the form of ideal gas equation of state, with the pressure proportional to the density. We investigate the collapse/expansion of condensed dark matter halos for this model, and we study the analytical/semi-analytical solutions of the hydrodynamic evolution equations, derived by using the method of separation of variables. An approximate 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, and studied by using numerical methods.