  We study the formation and evolution of solitons supported by repulsive
self-interactions inside extended halos, for scalar-field dark matter
scenarios. We focus on the semiclassical regime where the quantum pressure is
typically much smaller than the self-interactions. We present numerical
simulations, with initial conditions where the halo is described by the WKB
approximation for its eigenfunction coefficients. We find that when the size of
the system is of the order of the Jeans length associated with the
self-interactions, a central soliton quickly forms and makes about 50% of the
total mass. However, if the halo is ten times greater than this
self-interaction scale, a soliton only quickly forms in cuspy halos where the
central density is large enough to trigger the self-interactions. If the halo
has a flat core, it takes a longer time for a soliton to appear, after small
random fluctuations on the de Broglie wavelength size build up to reach a large
enough density. In some cases, we observe the co-existence of several narrow
density spikes inside the larger self-interaction-supported soliton. All
solitons appear robust and slowly grow, unless they already make up 40% of the
total mass. We develop a kinetic theory, valid for an inhomogeneous background,
to estimate the soliton growth rate for low masses. It explains the fast
falloff of the growth rate as resonances between the ground state and halo
excited states disappear. Our results suggest that cosmological halos would
show a large scatter for their soliton mass, depending on their assembly
history.

Cosmology and Nongalactic Astrophysics