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We investigate nonlinear structure formation in the fuzzy dark matter (FDM) model using both numerical and perturbative techniques. On the numerical side, we examine the virtues and limitations of a Schr\"odinger-Poisson solver (wave formulation) versus a fluid dynamics solver (Madelung formulation). On the perturbative side, we carry out a computation of the one-loop mass power spectrum, i. e. up to third order in perturbation theory. We find that (1) in many situations, the fluid dynamics solver is capable of producing the expected interference patterns, but it fails in situations where destructive interference causes the density to vanish---a generic occurrence in the nonlinear regime. (2) The Schr\"odinger-Poisson solver works well in all test cases, but it is demanding in resolution: suppose one is interested in the mass power spectrum on large scales, it is not sufficient to resolve structure on those same scales; one must resolve the relevant de Broglie scale which is often smaller. The fluid formulation does not suffer from this issue. (3) We compare the one-loop mass power spectrum from perturbation theory against the mass power spectrum from the Schr\"odinger-Poisson solver, and find good agreement in the mildly nonlinear regime. We contrast fluid perturbation theory with wave perturbation theory; the latter has a more limited range of validity. (4) As an application, we compare the Lyman-alpha forest flux power spectrum obtained from the Schr\"odinger-Poisson solver versus one from an N-body simulation (the latter is often used as an approximate method to make predictions for FDM). At redshift 5, the two, starting from the same initial condition, agree to better than 10% on observationally relevant scales as long as the FDM mass exceeds 210^-23 eV. We emphasize that the so called quantum pressure is capable of both enhancing and suppressing fluctuations in the nonlinear regime---which dominates depends on the scale and quantity of interest.
Li et al. (Tue,) studied this question.