Nonlinear gravitational sound waves within an expanding collisionless background matter are investigated. Under the assumptions of power-law time dependence and an equation of state for collisionless particles, two classes of nonlinear waves are identified: solitons and solitary wave trains. The soliton solutions may describe physically how the high-density sheets evolve immediately after the formation of caustics in the disk-collapse scenario of large-scale structure formation. The solutions can describe either that a discoid continuously loses matter that is expelled by the high pressure of the disk, reaching supersonic speeds and presumably blended into the background Hubble flow, or that the disk collects matter from the low-density background and becomes ever-increasingly massive. We also find that the multiple-mass-shell solutions, generally believed to be caused by the crossing of mass shells after the gravitational collapse, exist in the form of solitons.