  We employ isolated N-body simulations to study the response of
self-interacting dark matter (SIDM) halos in the presence of the baryonic
potentials. Dark matter self-interactions lead to kinematic thermalization in
the inner halo, resulting in a tight correlation between the dark matter and
baryon distributions. A deep baryonic potential shortens the phase of SIDM core
expansion and triggers core contraction. This effect can be further enhanced by
a large self-scattering cross section. We find the final SIDM density profile
is sensitive to the baryonic concentration and the strength of dark matter
self-interactions. Assuming a spherical initial halo, we also study evolution
of the SIDM halo shape together with the density profile. The halo shape at
later epochs deviates from spherical symmetry due to the influence of the
non-spherical disc potential, and its significance depends on the baryonic
contribution to the total gravitational potential, relative to the dark matter
one. In addition, we construct a multi-component model for the Milky Way,
including an SIDM halo, a stellar disc and a bulge, and show it is consistent
with observations from stellar kinematics and streams.
