Nonlinear instability and solitons in a self-gravitating fluid

We study a spherical, self-gravitating fluid model, which finds applications in cosmic structure formation. We argue that since the system features nonlinearity and gravity-induced dispersion, the emergence of solitons becomes possible. We thus employ a multiscale expansion method to study, in the weakly nonlinear regime, the evolution of small-amplitude perturbations around the equilibrium state. This way, we derive a spherical nonlinear Schr{\"o}dinger (NLS) equation that governs the envelope of the perturbations. The effective NLS description allows us to predict a "nonlinear instability" (occurring in the nonlinear regime of the system), namely, the modulational instability which, in turn, may give rise to spherical soliton states. The latter feature a very slow (polynomial) curvature-induced decay in time. The soliton profiles may be used to describe the shape of dark matter halos at the rims of the galaxies.