Single and merger soliton dynamics in scalar field dark matter with and without self-interactions

(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 and make our amended code public. 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 postmerger objects, and fit them to analytical functions of previous literature. We find a mild preference for Gaussian cores for objects with SI, while the envelopes of postmergers 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 postmergers, confirming that self-gravitating SFDM has very long relaxation times, although objects with SI exhibit oscillations of comparatively smaller amplitude.