The dust disk dynamics in week-nonlinear regime
  We investigate the problem of self-gravitating dust disk dynamics in a static state taking into account nonlinear effects. For this purpose Schrodinger-type equation including the mass conservation law is used for the whole description of hydrodynamic flows of self-gravitating dust. We have shown a purely hydrodynamic mechanism of ring formation in the radial direction by taking into account nonlinearity in the lowest order of expansion parameter, which determines an order of magnitude of flow. 
Victor M. Zhuravlev, Alexander V. Patrushev