Nonlinear disturbances in self-gravitating media

Using a multiple time-scale method, the weakly nonlinear waves on a self-gravitating incompressible fluid column are investigated. The analysis reveals that near the wavenumberk=k_c, the amplitude modulation of a standing wave can be described by the nonlinear Schrödinger equation with the roles of time and space variables interchanged. The nonlinear cutoff wavenumber, which depends sensitively on initial conditions, can then be derived from the nonlinear Schrödinger equation so obtained. The finite amplitude standing wave is stable against modulation.