Decay of shocklike waves into compact oscillons

The signum-Gordon model in 1 + 1 dimensions possesses an exact solution with a discontinuity of the field at the light cone. Thus, the gradient energy of these waves is infinite whereas the energy of a part of the wave localized inside the light cone is finite and grows linearly with time. The initial data for such waves contain a field configuration which is null in space and its time derivative is proportional to the Dirac delta. We study regular initial data that lead to shocklike waves with finite gradient energy. We found that such waves exist only for a finite time and eventually decay in a cascade of oscillonlike structures. The pattern of decay is very similar to the one observed in the scattering of compact oscillons.