Integrable models of the dynamics of a compressible medium in its own gravitational field. The Cole – Hopf substitution method

The exact solutions of the self-coordinated problem of dynamics of the compressible fluid in self-gravitational field are developed by the authors on the basis of modifications to the Cole – Hopf generalized substitutions method. The article considers various models of a one-dimensional flow of the ideal and viscous self-gravitating fluid for the general view of initial distributions of its density and velosity. The authors calculate the integrals of movement and generalized Cole – Hopf substitutions, allowing to construct exact decisions. The method of construction of exact solutions is generalized for the ideal fluid in self-gravitational field on the two-dimensional and three-dimensional flows possessing cylindrical and spherical simmetry. The article analyzes the processes of formation of dense objects from a dust cloud with the set initial distribution of density of dust owing to Jeanse's instability for flat and spherical simmetry on the basis of the constructed decisions. The researchers also calculate the characteristic time before singularity formation.