Numerical simulation of gravitational instability of the Sun protoplanetary disk in the one-dimensional approximation. Part I. A homogeneous isotropic medium

There were considered the analytical and numerical solutions of the motion equations of a homogeneous isotropic infinite gravitating gaseous medium in the two approximations: “cold” gas and gas at the final temperature.
There were obtained real solutions, describing the behavior of a uniform medium wave disturbances, and single disturbances. Waves of gravitational instability, the amplitude of which is growing exponentially, and the highs and lows of this wave, as well as its nodal points, retain its position in space, follow the basic laws of Jeans model. The authors interpret this wave of instability as an analogue protoplanetary rings that can be formed in protoplanetary disks.
According to the results of numerical calculations homogeneous gravitating medium reaction to the initial localized (single) perturbation of its density is significantly different from the laws of Jeans model. Instability localized initial perturbations extends to the region $\lambda<\lambda_J$, although in this case the growth of density perturbations is considerably less than when $\lambda>\lambda_J$.
It was found that the gravitational instability in the region $\lambda>\lambda_J$ suppress sound.
It is shown that without taking into account the rotation of the medium of the Sun protoplanetary disk its critical density in the event of a large-scale gravitational instability to four orders of magnitude is less than the critical density, obtained in the framework of the theory of formation of planets by accumulation of solids and particles.