LAUSR

We show that equations of Newtonian hydrodynamics and gravity describing one-dimensional steady gas flow possess nonlinear periodic solutions. In the case of a zero-pressure gas the solution exhibits hydrodynamic similarity and is universal: it is a lattice of integrable density singularities coinciding with maxima of the gravitational potential. With finite pressure effects included, there exists critical matter density that separates two regimes of behavior. If the average density is below the critical, the solution is a density wave which is in phase with the wave of the gravitational potential. If the average density is above the critical, the waves of the density and potential are out of phase. Traveling nonlinear gravitostatic waves are also predicted and their properties elucidated. Specifically, subsonic wave is made out of two out of phase oscillations of matter density and gravitational potential. If the wave is supersonic, the density-potential oscillations are in phase.