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Peakons are singular, soliton-like solutions to nonlinear wave equationswhose dynamics can be studied using ordinary differential equations (ODEs). TheDegasperis-Procesi equation (DP) is an important example of an integrable PDEexhibiting wave breaking in the peakon sector thus affording an interpretationof wave breaking as a mechanical collision of particles. In this paper we setup a general formalism in which to study collisions of DP peakons and apply it,as an illustration, to a detailed study of three colliding peakons. It is shownthat peakons can collide only in pairs, no triple collisions are allowed and atthe collision a shockpeakon is created. We also show that the initialconfiguration of peakon-antipeakon pairs is nontrivially correlated with thespectral properties of an accompanying non-selfadjoint boundary value problem.In particular if peakons or antipeakons are bunched up on one side relative toa remaining antipeakon or peakon then the spectrum is real and simple. Eventhough the spectrum is in general complex the existence of a global solution ineither time direction dynamics is shown to imply the reality of the spectrum ofthe boundary value problem.

Peakon-antipeakon interactions in the Degasperis-Procesi Equation