  We present the first study on the gravitational impact of supernova feedback
in an isolated soliton and a spherically symmetric dwarf SFDM halo of virial
mass $1\times 10^{10}\mathrm{M_\odot}$. We use a boson mass
$m=10^{-22}\mathrm{eV/c^2}$ and a soliton core $r_c \approx 0.7$kpc, comparable
to typical half-light radii of Local Group dwarf galaxies. We simulate the
rapid gas removal from the center of the soliton by a concentric external
time-dependent Hernquist potential. We explore two scenarios of feedback
blowouts: i) a massive single burst, and ii) multiple consecutive blowouts
injecting the same total energy to the system, including various magnitudes for
the blowouts in both scenarios. In all cases, we find one single blowout has a
stronger effect on reducing the soliton central density. Feedback leads to
central soliton densities that oscillate quasi-periodically for an isolated
soliton and stochastically for a SFDM halo. The range in the density amplitude
depends on the strength of the blowout, however we observe typical variations
of a factor of $\geqslant$2. One important consequence of the stochastic
fluctuating densities is that, if we had no prior knowledge of the system
evolution, we can only know the configuration profile at a specific time within
some accuracy. By fitting soliton profiles at different times to our simulated
structures, we found the (1-$\sigma$) scatter of their time-dependent density
profiles. For configurations within the 1$\sigma$ range, we find the inferred
boson mass is typically less than 20\% different from the real value used in
our simulations. Finally, we compare the observed dynamical masses of field
dwarf galaxies in our Local Group with the implied range of viable solitons
from our simulations and find good agreement.
