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

Scalar Field Dark Matter: Impact of Supernovae-driven blowouts on the soliton structure of low mass dark matter halos