  We consider the scenario of self-interacting dark matter(SIDM) with a light
mediator in a model-independent way, which can alleviate two long-standing
issues of the small scale cosmology namely cusp vs. core and too-big-to-fail. A
Yukawa potential is chosen to achieve mediator exchange between DM particles as
part of their self-interactions. The dynamics of self-interacting transfer
cross-section are studied for a range of mediator mass($m_Z'$). Also, a
relationship is established between the cross-section and DM particles'
relative velocity, which ensures the solution to the DM crisis at small scales.
Our obtained numerical results are efficient compared to the earlier works in
the context that a lesser number of $\ell$ modes have been used by us to
achieve the same level of accuracy in the cross-section calculations. For a
better understanding of the SIDM parameter space, we perform an analytical
analysis on the dependence of transfer cross-section over the other important
SIDM parameters using a Hulth\'{e}n potential which is similar in its behavior
to Yukawa potential. A detailed evolution of particle dynamics using the
Boltzmann equation and the effect of Sommerfeld enhancement on such
calculations has also been studied here. We also provide a minimal anomaly-free
leptophilic extension of the standard model, that can incorporate SIDM and its
mediator candidate in the framework.
