Core-Envelope Haloes in Scalar Field Dark Matter with Repulsive
  Self-Interaction: Fluid Dynamics Beyond the de Broglie Wavelength
  Scalar Field Dark Matter (SFDM) comprised of ultralight bosons has attracted
great interest as an alternative to standard, collisionless Cold Dark Matter
(CDM) because of its novel structure-formation dynamics, described by the
coupled Schr\"odinger-Poisson equations. In the free-field ("fuzzy") limit of
SFDM (FDM), structure is inhibited below the de Broglie wavelength, but
resembles CDM on larger scales. Virialized haloes have "solitonic" cores of
radius $\sim\lambda_\text{deB}$, surrounded by CDM-like envelopes. When a
strong enough repulsive self-interaction (SI) is also present, structure can be
inhibited below a second length scale, $\lambda_\text{SI}$, with
$\lambda_\text{SI}> \lambda_\text{deB}$ -- called the Thomas-Fermi (TF) regime.
FDM dynamics differs from CDM because of quantum pressure, and SFDM-TF differs
further by adding SI pressure. In the small-$\lambda_\text{deB}$ limit,
however, we can model all three by fluid conservation equations for a
compressible, $\gamma=5/3$ ideal gas, with ideal gas pressure sourced by
internal velocity dispersion and, for the TF regime, an added SI pressure,
$P_\text{SI}\propto \rho^2$. We use these fluid equations to simulate halo
formation from gravitational collapse in 1D, spherical symmetry, demonstrating
for the first time that SFDM-TF haloes form with cores the size of
$R_\text{TF}$, the radius of an SI-pressure-supported $(n=1)$-polytrope,
surrounded by CDM-like envelopes. In comparison with rotation curves of dwarf
galaxies in the local Universe, SFDM-TF haloes pass the ["too-big-to-fail" +
"cusp-core"]-test if $R_\text{TF}\gtrsim 1$ kpc.
