Solitons in Newtonian gravity

It is shown that the plane-wave solutions for the equations governing the motion of a self-gravitating isothermal fluid in Newtonian hydrodynamics are generated by a sine-Gordon equation which is solvable by an 'inverse scattering' transformation. A transformation procedure is outlined by means of which one can construct solutions of the gravity system (2) out of a pair of solutions of the sine-Gordon equation which are interrelated via an auto-Backlund transformation. In general the solutions to the gravity system (2) are obtained in a parametric representation in terms of characteristic coordinates. All solutions of (2) generated by the one- and two-soliton solutions of the sine-Gordon equation can be constructed explicitly. These might provide models for the evolution of flat structures as they are predicted to arise in the process of galaxy formation.