  In scalar-field dark matter scenarios, a scalar-field soliton could form at
the center of galactic halos, around the supermassive black holes that sit at
the center of galaxies. Focusing on the large scalar-mass limit, where the
soliton is formed by the balance between self-gravity and a repulsive
self-interaction, we study the infall of the scalar field onto the central
Schwarzschild black hole. We derive the scalar-field profile, from the
Schwarzschild radius to the large radii dominated by the scalar cloud. We show
that the steady state solution selects the maximum allowed flux, with a
critical profile that is similar to the transonic solution obtained for the
hydrodynamic case. This finite flux, which scales as the inverse of the
self-interaction coupling, is small enough to allow the dark matter soliton to
survive for many Hubble times.
