  Wave dark matter ($\psi$DM), which satisfies the Schr\"odinger-Poisson
equation, has recently attracted substantial attention as a possible dark
matter candidate. Numerical simulations have in the past provided a powerful
tool to explore this new territory of possibility. Despite their successes to
reveal several key features of $\psi$DM, further progress in simulations is
limited, in that cosmological simulations so far can only address formation of
halos below $\sim 2\times 10^{11} M_\odot$ and substantially more massive halos
have become computationally very challenging to obtain. For this reason, the
present work adopts a different approach in assessing massive halos by
constructing wave-halo solutions directly from the wave distribution function.
This approach bears certain similarity with the analytical construction of
particle-halo (cold dark matter model). Instead of many collisionless
particles, one deals with one single wave that has many non-interacting
eigenstates. The key ingredient in the wave-halo construction is the
distribution function of the wave power, and we use several halos produced by
structure formation simulations as templates to determine the wave distribution
function. Among different models, we find the fermionic King model presents the
best fits and we use it for our wave-halo construction. We have devised an
iteration method for constructing the nonlinear halo, and demonstrate its
stability by three-dimensional simulations. A Milky-Way-sized halo has also
been constructed, and the inner halo is found flatter than the NFW profile.
These wave-halos have small-scale interferences both in space and time
producing time-dependent granules. While the spatial scale of granules varies
little, the correlation time is found to increase with radius by one order of
magnitude across the halo.
