  We apply spectral stability theory to investigate nonlinear gravity waves in
the atmosphere. These waves are determined by modulation equations that result
from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves,
which represent exact solutions to the inviscid Boussinesq equations, are
spectrally stable with respect to their nonlinear modulation equations under
the same conditions as what is known as modulational stability from weakly
nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime
does account for the altitudinal varying background density. Second, we show
for the first time that upward-traveling wave fronts solving the inviscid
modulation equations, that compare to pseudo-incompressible theory, are
unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the
spectra allow for arbitrarily large instability growth rates. Third, a
regularization is found by including dissipative effects. The corresponding
traveling wave solutions have localized amplitude and blow up unconditionally
by embedded eigenvalue instabilities but the instability growth rate is bounded
from above. Additionally, all three types of nonlinear modulation equations are
solved numerically to further investigate and illustrate the nature of the
analytic stability results.
