  (abridged)Scalar field dark matter (SFDM) made of bosons has become a popular
alternative to the CDM paradigm, especially for its potential to cure the
so-called "small-scale problems" of CDM. Cosmological simulations have
determined that SFDM halos exhibit a core-envelope structure, but they are
computationally expensive. Halo cores have been found to be well approximated
by "solitons". The study of single soliton and multiple soliton merger dynamics
constitutes a more feasible approach to investigate in detail the genuine
quantum dynamics of SFDM and its interplay with self-gravity for a multitude of
free boson parameters. In this paper, we present dedicated simulations of
single solitons and binary soliton mergers, for models without and with a
2-boson, repulsive, weak to intermediate self-interaction (SI), as well as
multiple soliton mergers without SI. We adapt the open-source code Pyultralight
to simulate solitons with SI. We derive numerical scaling relations between the
central density and mass of solitons for several values of SI and find
deviations from the monotonic relations known from fuzzy dark matter (no SI),
or the strongly repulsive Thomas-Fermi regime. Solitons with SI exemplify
larger cores and lower central densities, compared to solitons without SI.
Using our simulations, we extract numerical density profiles for solitons and
post-merger objects, and fit them to analytic functions of previous literature.
We find a mild preference for Gaussian cores for objects with SI, while the
envelopes of post-mergers can be fit to NFW profiles albeit with some caution
as we discuss. Similar to previous work, we find global, persistent
oscillations for solitons as well as post-mergers, confirming that
self-gravitating SFDM has very long relaxation times, although objects with SI
exhibit oscillations of comparatively smaller amplitude.
