  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 Schrodinger-Poisson solver (wave
formulation) versus a fluid dynamics solver (Madelung formulation). We also
carry out a perturbative computation of the one-loop mass power spectrum. We
find that (1) in many cases, the fluid dynamics solver is capable of producing
the expected interference patterns, but it fails where destructive interference
causes the density to vanish which generally occurs in the nonlinear regime.
(2) The Schrodinger-Poisson solver works well in all test cases, but it is
demanding in resolution: one must resolve the small de Broglie scale to obtain
the correct dynamics on large scales. (3) We compare the mass power spectrum
from perturbation theory against that from the Schrodinger-Poisson solver, and
find good agreement in the mildly nonlinear regime. Compared with fluid
perturbation theory, wave perturbation theory has a more limited range of
validity. (4) As an application, we compare the Lyman-alpha forest flux power
spectrum obtained from the Schrodinger-Poisson solver versus one from an N-body
simulation (which 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 $2 \times 10^{-23}$ eV.
