Shock creation and Painlev\'{e} property of colliding peakons in the Degasperis-Procesi Equation

Abstract

The Degasperis-Procesi equation (DP) is one of several equations known to model important nonlinear effects such as wave breaking and shock creation. It is, however, a special property of the DP equation that these two effects can be studied in an explicit way with the help of the multipeakon ansatz. In essence this ansatz allows one to model wave breaking as a collision of hypothetical particles (peakons and antipeakons), called henceforth collectively multipeakons. It is shown that DP multipeakons have Painlev\'{e} property which implies a universal wave breaking behaviour, that multipeakons can collide only in pairs, and that there are no multiple collisions other than, possibly simultaneous, collisions of peakon-antipeakon pairs at different locations. Moreover, it is demonstrated that each peakon-antipeakon collision results in creation of a shock thus making possible a multi shock phenomenon.