We study solutions close to solitary waves of the pseudo-relativistic Hartree equation describing boson stars under the influence of an external gravitational field. In particular, we analyze the long-time effective dynamics of such solutions. In essence, we establish a (long-time) stability result for solutions describing boson stars that move under the influence of an external gravitational field. The proof of our main result tackles difficulties that are absent when deriving similar results on effective solitary wave motions for NLS or NLW. This is due to the fact that the pseudorelativisitic Hartree equation does not exhibit Galilean or Lorentz covariance.
This is a joint work with J. Fröhlich, Theoretical Physics, ETH Zurich and E. Lenzmann, Department of Mathematics, MIT.