  Fuzzy Dark Matter (FDM) represents an alternative and intriguing description
of the standard Cold Dark Matter (CDM) fluid, able to explain the lack of
direct detection of dark matter particles in the GeV sector and to alleviate
small scales tensions in the cosmic large-scale structure formation.
Cosmological simulations of FDM models in the literature were performed either
with very expensive high-resolution grid-based simulations of individual haloes
or through N-body simulations encompassing larger cosmic volumes but resorting
on significant approximations in the FDM non-linear dynamics to reduce their
computational cost. With the use of the new N-body cosmological hydrodynamical
code AX-GADGET, we are now able not only to overcome such numerical problems,
but also to combine a fully consistent treatment of FDM dynamics with the
presence of gas particles and baryonic physical processes, in order to quantify
the FDM impact on specific astrophysical observables. In particular, in this
paper we perform and analyse several hydrodynamical simulations in order to
constrain the FDM mass by quantifying the impact of FDM on Lyman-$\alpha$
forest observations, as obtained for the first time in the literature in a
N-body setup without approximating the FDM dynamics. We also study the
statistical properties of haloes, exploiting the large available sample, to
extract information on how FDM affects the abundance, the shape, and density
profiles of dark matter haloes.
