Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation

Solutions of the Degasperis-Procesi nonlinear wave equation may develop discontinuities in finite time. As shown by Coclite and Karlsen, there is a uniquely determined entropy weak solution which provides a natural continuation of the solution past such a point. Here we study this phenomenon in detail for solutions involving interacting peakons and antipeakons. We show that a jump discontinuity forms when a peakon collides with an antipeakon, and that the entropy weak solution in this case is described by a "shockpeakon" ansatz reducing the PDE to a system of ODEs for positions, momenta, and shock strengths.