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 $2times10^{12} {M_odot}$ at $z=8$ should have massive solitonic cores of $sim 2times10^9 {M_odot}$ 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.
