  We study the formation and evolution of self-interacting dark matter (SIDM)
halos. We find analytical, fully cosmological similarity solutions taking
account of the collisional interaction of SIDM particles. This interaction
results in a thermal conductivity that heats the halo core and flattens its
density profile. These similarity solutions are relevant to galactic and
cluster halo formation in the CDM model. We assume an initial mass profile dM/M
M^{-eps}, as in the familiar secondary infall model. If eps=1/6, SIDM halos
will evolve self-similarly, with a cold, supersonic infall terminated by a
strong accretion shock. Different solutions arise for different values of the
collisionality parameter, Q= sigma rho_b r_s, where sigma is the scattering
cross section, rho_b is the cosmic mean density, and r_s is the shock radius.
For all these solutions, a flat-density, isothermal core is present which grows
in size as a fixed fraction of r_s. We find two different regimes for these
solutions: 1) for Q \leq Q_{th}, the core density decreases and core size
increases as Q increases; 2) for Q \geq Q_{th}, the core density increases and
core size decreases as Q increases. Our similarity solutions are in agreement
with previous N-body simulations of SIDM halos, which correspond to the low-Q
regime, if Q=[8.4e-4 - 4.9e-2]Q_{th} (low-Q), or sigma=[0.56-5.6]cm^2/g. As
Q=\infty, our similarity solution aquires a central density cusp, in agreement
with some simulation results which used an ordinary collisional fluid to
approximate the effects of SIDM collisionality. When Q=[18.6-231]Q_{th} or
sigma=[1.2e4 - 2.71e4]cm^2/g, for which we find flat-density cores comparable
to those of the observationally acceptable low-Q solutions, has not previously
been identified. Further study of this regime is warranted.
