Patterns of Gravitational Cooling in Schrodinger Newton System
  We study time evolution of Schrodinger-Newton system using the
self-consistent Crank-Nicolson method to understand the dynamical
characteristics of nonlinear systems. Compactifying the radial coordinate by a
new one, which brings the spatial infinity to a finite value, we are able to
impose the boundary condition at infinity allowing for a numerically exact
treatment of the Schrodinger-Newton equation. We study patterns of
gravitational cooling starting from exponentially localized initial states.
When the gravitational attraction is strong enough, we find that a small-sized
oscillatory solitonic core is forming quickly, which is surrounded by a growing
number of temporary halo states. In addition a significant fraction of
particles escape to asymptotic regions. The system eventually settles down to a
stable solitonic core state while all the excess kinetic energy is carried away
by the escaping particles, which is a phenomenon of gravitational cooling.
