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One of the hottest questions in the cosmology of self-interacting dark matter
(SIDM) is whether scatterings can induce detectable core-collapse in halos by
the present day. Because gravitational tides can accelerate core-collapse, the
most promising targets to observe core-collapse are satellite galaxies and
subhalo systems. However, simulating small subhalos is computationally
intensive, especially when subhalos start to core-collapse. In this work, we
present a hierarchical framework for simulating a population of SIDM subhalos,
which reduces the computation time to linear order in the total number of
subhalos. With this method, we simulate substructure lensing systems with
multiple velocity-dependent SIDM models, and show how subhalo evolution depends
on the SIDM model, subhalo mass and orbits. We find that an SIDM cross section
of $\gtrsim 200$ cm$^2$/g at velocity scales relevant for subhalos' internal
heat transfer is needed for a significant fraction of subhalos to core-collapse
in a typical lens system at redshift $z=0.5$, and that core-collapse has unique
observable features in lensing. We show quantitatively that core-collapse in
subhalos is typically accelerated compared to field halos, except when the SIDM
cross section is non-negligible ($\gtrsim \mathcal{O}(1)$ cm$^2$/g) at
subhalos' orbital velocities, in which case evaporation by the host can delay
core-collapse. This suggests that substructure lensing can be used to probe
velocity-dependent SIDM models, especially if line-of-sight structures (field
halos) can be distinguished from lens-plane subhalos. Intriguingly, we find
that core-collapse in subhalos can explain the recently reported ultra-steep
density profiles of substructures found by lensing with the \emph{Hubble Space
Telescope}