Formation of solitons and their transitions in scalar-field dark matter models with a nonpolynomial self-interaction potential

We study the formation of solitons inside scalar-field dark matter halos with a nonpolynomial self-interaction potential. We consider a self-interaction potential that is quartic in the scalar field in the low-density regime but saturates at large densities. This mimics the behavior of axion monodromy potentials. We concentrate on the semiclassical regime, where the de Broglie wavelength is much smaller than the size of the system. We find that depending on the strength and scale of the self-interactions, the system can form solitons of the Thomas-Fermi type (dominated by self-interactions) or of the fuzzy dark matter type (dominated by the quantum pressure). The system can also display transitions from a Thomas-Fermi soliton to a fuzzy dark matter soliton as the former becomes unstable. We show that these behaviors can be understood from a simple Gaussian ansatz. We find that even in cases where the self-interactions are always subdominant they can play a critical role, by providing a small density boost that is enough to generate the seed for the formation of a fuzzy dark matter soliton at much later times. We also point out that the intuition derived from a hydrodynamical picture can be misleading in regimes where wave effects are important.