  The scale-free gravothermal fluid formalism has long proved effective in
describing the evolution of self-interacting dark matter halos with a constant
dark matter particle cross section. However, whether the gravothermal fluid
solutions match numerical simulations for velocity-dependent cross-section
scenarios remains untested. In this work, we provide a fast mapping method that
relates the constant-cross-section gravothermal solution to models with
arbitrary velocity dependence in the cross section. We show that the
gravothermal solutions after mapping are in good agreement with Arepo N-Body
simulation results. We illustrate the power of this approach by applying this
fast mapping method to a halo hosting a low surface brightness galaxy UGC 128.
We show that this fast mapping method can be used to constrain free parameters
in a physically motivated cross-section model and illustrate parameter space
favored by the rotation curve measurement.
