  We present non-linear solutions of Vlasov Perturbation Theory (VPT),
describing gravitational clustering of collisionless dark matter with
dispersion and higher cumulants induced by orbit crossing. We show that VPT can
be cast into a form that is formally analogous to standard perturbation theory
(SPT), but including additional perturbation variables, non-linear
interactions, and a more complex propagation. VPT non-linear kernels have a
crucial decoupling property: for fixed total momentum, the kernels becomes
strongly suppressed when any of the individual momenta cross the dispersion
scale into the non-linear regime. This screening of UV modes allows us to
compute non-linear corrections to power spectra even for cosmologies with very
blue power-law input spectra, for which SPT diverges. We compare predictions
for the density and velocity divergence power spectra as well as the bispectrum
at one-loop order to N-body results in a scaling universe with spectral indices
$-1\leq n_s\leq +2$. We find a good agreement up to the non-linear scale for
all cases, with a reach that increases with the spectral index $n_s$. We
discuss the generation of vorticity as well as vector and tensor modes of the
velocity dispersion, showing that neglecting vorticity when including
dispersion would lead to a violation of momentum conservation. We verify
momentum conservation when including vorticity, and compute the vorticity power
spectrum at two-loop order, necessary to recover the correct large-scale limit
with slope $n_w=2$. Comparing to our N-body measurements confirms the
cross-over from $k^4$ to $k^2$ scaling on large scales. Our results provide a
proof-of-principle that perturbative techniques for dark matter clustering can
be systematically improved based on the known underlying collisionless
dynamics.
