Self-gravitating fluid dynamics, instabilities, and solitons

This work studies the hydrodynamics of self-gravitating, compressible, isothermal fluids. We show that the hydrodynamic-evolution equations are scale covariant in the absence of viscosity. Then, we study the evolution of the time-dependent fluctuations around singular and regular isothermal spheres. We linearize the fluid equations around such stationary solutions and develop a method based on the Laplace transform to analyze their dynamical stability. We find that the system is stable below a critical size $(X\sim 9.0$ in dimensionless variables) and unstable above; this criterion is the same as the one found for the thermodynamic stability in the canonical ensemble and it is associated with a center-to-border density ratio of 32.1. We prove that the value of this critical size is independent of the Reynolds number of the system. Furthermore, we give a detailed description of the series of successive dynamic instabilities that appear at larger and larger sizes following the geometric progression $X_n\sim {10.7}^n,$ $n=1,2,\dots .$ Then, we search for exact solutions of the hydrodynamic equations without viscosity, we provide analytic and numerical axisymmetric soliton-type solutions. The stability of exact solutions corresponding to a collapsing filament is studied by computing linear fluctuations. Radial fluctuations growing faster than the background are found for all sizes of the system. However, a critical size $(X\sim 4.5)$ appears, separating a weakly from a strongly unstable regime.