Nonlinear Waves in a Self‐Gravitating Medium

Abstract

To investigate the effects of finite amplitudes on the behavior of waves in a self‐gravitating medium, exact solutions of nonlinear waves are obtained in a simple situation. The waves are traveling in the direction of the axis of rotation of a homogeneous and uniformly rotating self‐gravitating medium. The nonlinear wave profile and the dispersion relation are obtained and compared with Jeans's linear theory. It is found that the wavelength‐amplitude plane can be divided into three regions: (1) a region where the shock wave is expected to form, (2) a region where the neutral wave exists, and (3) a region where there is gravitational instability. In the third case, it is expected that the unstable wave will eventually evolve to a neutral wave with finite amplitude. Based on this result, speculations are made on the evolution of the unstable density wave in a galaxy. It is established that the propagation of the nonlinear wave in a self‐gravitating medium is always “subsonic,” just as in the linear theory. The difference is that the maximum ratio of the phase speed to the sound speed of the basic state is 1 − A2 for the isothermal case and (1 − A2)(γ + 1)/2 for the polytropic case instead of 1 for the linear case, where A2 is the dimensionless depth of the valley of the density profile, and γ is the polytropic index. It is also shown that no solitary wave can be found in a homogeneous self‐gravitating medium. Comparison is made with nonlinear waves in a homogeneous plasma.