  We examine the nonlinear structure of gravitationally collapsed objects that
form in our simulations of wavelike cold dark matter ($\psi$DM), described by
the Schr\"{o}dinger-Poisson (SP) equation with a particle mass $\sim 10^{-22}
{\rm eV}$. A distinct gravitationally self-bound solitonic core is found at the
center of every halo, with a profile quite different from cores modeled in the
warm or self-interacting dark matter scenarios. Furthermore, we show that each
solitonic core is surrounded by an extended halo composed of large fluctuating
dark matter granules which modulate the halo density on a scale comparable to
the diameter of the solitonic core. The scaling symmetry of the SP equation and
the uncertainty principle tightly relate the core mass to the halo specific
energy, which, in the context of cosmological structure formation, leads to a
simple scaling between core mass ($M_c$) and halo mass ($M_h$), $M_c \propto
a^{-1/2} M_h^{1/3}$, where $a$ is the cosmic scale factor. We verify this
scaling relation by (i) examining the internal structure of a statistical
sample of virialized halos that form in our 3D cosmological simulations, and by
(ii) merging multiple solitons to create individual virialized objects.
Sufficient simulation resolution is achieved by adaptive mesh refinement and
graphic processing units acceleration. From this scaling relation, present
dwarf satellite galaxies are predicted to have kpc sized cores and a minimum
mass of $\sim 10^8 {M_\odot}$, capable of solving the small-scale controversies
in the cold dark matter model. Moreover, galaxies of $2\times10^{12} {M_\odot}$
at $z=8$ should have massive solitonic cores of $\sim 2\times10^9 {M_\odot}$
within $\sim 60 {\rm pc}$. Such cores can provide a favorable local environment
for funneling the gas that leads to the prompt formation of early stellar
spheroids and quasars.
