  Fuzzy Dark Matter (FDM) models admit self-similar solutions which are very
different from their Cold Dark Matter (CDM) counterparts and do not converge to
the latter in the semiclassical limit. In contrast with the familiar CDM
hierarchical collapse, they correspond to an inverse-hierarchy blow-up.
Constant-mass shells start in the nonlinear regime, at early times, with small
radii and high densities, and expand to reach at late times the Hubble flow, up
to small linear perturbations. Thus, larger masses become linear first. This
blow-up approximately follows the Hubble expansion, so that the central density
contrast remains constant with time, although the width of the self-similar
profile shrinks in comoving coordinates. As in a gravitational cooling process,
matter is ejected from the central peaks through successive clumps. As in wave
systems, the velocities of the geometrical structures and of the matter do not
coincide, and matter slowly moves from one clump to the next, with intermittent
velocity bursts at the transitions. These features are best observed using the
density-velocity representation of the nonrelativistic scalar field, or the
mass-shell trajectories, than with the Husimi phase-space distribution, where
an analogue of the Heisenberg uncertainty principle blurs the resolution in the
position or velocity direction. These behaviours are due to the quantum
pressure and the wavelike properties of the Schr\"odinger equation. Although
the latter has been used as an alternative to N-body simulations for CDM, these
self-similar solutions show that the semiclassical limit needs to be handled
with care.
