  We investigate the response of self-interacting dark matter (SIDM) halos to
the growth of galaxy potentials using idealized simulations, each run in tandem
with standard collisionless Cold Dark Matter (CDM). We find a greater diversity
in the SIDM halo profiles compared to the CDM halo profiles. If the stellar
gravitational potential strongly dominates in the central parts of a galaxy,
then SIDM halos can be as dense as CDM halos on observable scales. For extreme
cases with highly compact disks core collapse can occur, leading to SIDM halos
that are denser and cuspier than their CDM counterparts. If the stellar
potential is not dominant, then SIDM halos retain constant density cores with
densities far below CDM predictions. When a disk potential is present, the
inner SIDM halo becomes \em{more flattened} in the disk plane than the CDM
halo. These results are in excellent quantitative agreement with the
predictions of Kaplinghat et al. (2014). We also simulated a galaxy cluster
halo with a central stellar distribution similar to the brightest central
galaxy of the cluster A2667. A SIDM halo simulated with cross section over mass
$\sigma/m = 0.1\ \mathrm{cm^2 g^{-1}}$ provides a good match to the measured
dark matter density profile of A2667, while an adiabatically-contracted CDM
halo is denser and cuspier. The cored profile of the same halo simulated with
$\sigma/m = 0.5\ \mathrm{cm^2 g^{-1}}$ is not dense enough to match A2667. Our
findings are in agreement with previous results that $\sigma/m \gtrsim 0.1\
\mathrm{cm^2 g^{-1}}$ is disfavored for dark matter collision velocities in
excess of about 1500 km/s. More generally, the predictive cross-talk between
baryonic potentials and SIDM density distributions offers new directions for
constraining SIDM cross sections in massive galaxies where baryons are
dynamically important.
