  We study nonlinear wave phenomena in self-gravitating fluid systems, with a
particular emphasis on applications to molecular clouds. This paper presents
analytical results for one spatial dimension. We show that a large class of
physical systems can be described by theories with a ``charge density''
$q(\rho)$; this quantity replaces the density on the right hand side of the
Poisson equation for the gravitational potential. We use this formulation to
prove general results about nonlinear wave motions in self-gravitating systems.
We show that in order for stationary waves to exist, the total charge (the
integral of the charge density over the wave profile) must vanish. This
``no-charge'' property for solitary waves is related to the capability of a
system to be stable to gravitational perturbations for arbitrarily long
wavelengths. We find necessary and sufficient conditions on the charge density
for the existence of solitary waves and stationary waves. We then apply these
results to study nonlinear waves in Jeans-type theories, Yukawa theories, and a
pseudo-two-dimensional treatment. We study the allowed types of wave behavior
for all of these models. Finally, we discuss the implications of this work for
molecular cloud structure.
