  Fully 3D cosmological simulations of scalar field dark matter with
self-interactions, also known as Bose-Einstein condensate dark matter, are
performed using a set of effective hydrodynamic equations. These are derived
from the non-linear Schr\"odinger equation by performing a smoothing operation
over scales larger than the de Broglie wavelength, but smaller than the
self-interaction Jeans' length. The dynamics on the de Broglie scale become an
effective thermal energy in the hydrodynamic approximation, which is assumed to
be subdominant in the initial conditions, but become important as structures
collapse and the fluid is shock-heated. The halos that form have
Navarro-Frenk-White envelopes, while the centers become cored due to the fluid
pressures (thermal + self-interaction). The core radii are mostly determined by
the self-interaction Jeans' length, even though the effective thermal energy
eventually dominates over the self-interaction energy everywhere, a result that
is insensitive to the initial smallness of the thermal energy. Scaling
relations for the simulated population of halos are compared with Milky Way
dwarf spheroidals and nearby galaxies, assuming a Burkert halo profile, and are
found to not match, also for core radii $R_c\approx 1\text{kpc}$, which has
generally been the value expected to resolve the cusp-core issue. However, the
simulations have a limited volume, and therefore a limited halo mass range,
include no baryonic physics, and use fiducial cold dark matter initial
conditions with a cut-off near the Jeans' length at $z=50$, all of which can
affect the halo properties. [Abridged]
