Abstract
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.
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Lundmark, H. Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation. J Nonlinear Sci 17, 169–198 (2007). https://doi.org/10.1007/s00332-006-0803-3
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DOI: https://doi.org/10.1007/s00332-006-0803-3