Nonlinear perturbations of a rotating, self-gravitating gaseous disk

The possibility of self-organization of solitons into envelopes of spiral density waves in gaseous disks which are not required to be marginally stable gravitationally is examined theoretically. A governing differential equation is defined for a cubic nonlinearity in the perturbation amplitude according to the leading order of the WKB approximation. An expression is derived for the amplitude of the fundamental perturbation mode in the cubic approximation, thereby describing the behavior of the disturbances near an arbitrary wavenumber. A nonlinear dispersion relation is obtained for wave packet behavior in the disk and used to predict the wavenumber ranges which will accommodate formation of the solitons envelope. The wavenumbers are noted to be commensurate with presently accepted structural lengths of the spiral patterns in the Galaxy.