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We examine the nonlinear structure of gravitationally collapsed objects thatform in our simulations of wavelike cold dark matter ($\psi$DM), described bythe 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 thecenter of every halo, with a profile quite different from cores modeled in thewarm or self-interacting dark matter scenarios. Furthermore, we show that eachsolitonic core is surrounded by an extended halo composed of large fluctuatingdark matter granules which modulate the halo density on a scale comparable tothe diameter of the solitonic core. The scaling symmetry of the SP equation andthe uncertainty principle tightly relate the core mass to the halo specificenergy, which, in the context of cosmological structure formation, leads to asimple scaling between core mass ($M_c$) and halo mass ($M_h$), $M_c \proptoa^{-1/2} M_h^{1/3}$, where $a$ is the cosmic scale factor. We verify thisscaling relation by (i) examining the internal structure of a statisticalsample 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 andgraphic processing units acceleration. From this scaling relation, presentdwarf satellite galaxies are predicted to have kpc sized cores and a minimummass of $\sim 10^8 {M_\odot}$, capable of solving the small-scale controversiesin 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 environmentfor funneling the gas that leads to the prompt formation of early stellarspheroids and quasars.

Understanding the Core-Halo Relation of Quantum Wave Dark Matter, $\psi$DM, from 3D Simulations