  Cosmological information is usually extracted from the Lyman-$\alpha$ forest
correlations using only either large-scale information interpreted through
linear theory or using small-scale information interpreted by means of
expensive hydrodynamical simulations. A complete cosmological interpretation of
the 3D correlations at all measurable scales is challenged by the need of more
realistic models including the complex growth of non-linear small scales that
can only be studied within large hydrodynamical simulations. Past work were
often limited by the trade off between the simulated cosmological volume and
the resolution of the low-density intergalactic medium from which the
Lyman-$\alpha$ signal originates. We conduct a suite of hydrodynamical
simulations of the intergalactic medium, including one of the largest
Lyman-$\alpha$ simulations ever performed in terms of volume (640
$h^{-1}\mathrm{Mpc}$), alongside simulations in smaller volumes with
resolutions up to 25 $h^{-1}\mathrm{kpc}$. We compare the 3D Lyman-$\alpha$
power spectra predicted by those simulations to different non-linear models.
The inferred Lyman-$\alpha$ bias and RSD parameters, $b_\alpha$ and
$\beta_\alpha$ are in remarkable agreement with those measured in SDSS and DESI
data. We find that, contrary to intuition, the convergence of large-scale modes
of the 3D Lyman-$\alpha$ power spectra, which determines $\beta_\alpha$, is
primarily influenced by the resolution of the simulation box through mode
coupling, rather than the box size itself. Finally, we study the BAO signal
encoded in the 3D Lyman-$\alpha$ power spectra. For the first time with a
hydrodynamical simulation, we clearly detect the BAO signal, however we only
marginally detect its damping, associated with the non-linear growth of the
structures.
