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We derive explicit formulas for the characteristic curves associated with themultipeakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikovequations. Such a curve traces the path of a fluid particle whose instantaneousvelocity equals the elevation of the wave at that point (or the square of theelevation, in the Novikov case). The peakons themselves follow characteristiccurves, and the remaining characteristic curves can be viewed as paths of'ghostpeakons' with zero amplitude; hence, they can be obtained as solutions ofthe ODEs governing the dynamics of multipeakon solutions. The previously knownsolution formulas for multipeakons only cover the case when all amplitudes arenonzero, since they are based upon inverse spectral methods unable to detectthe ghostpeakons. We show how to overcome this problem by taking a suitablelimit in terms of spectral data, in order to force a selected peakon amplitudeto become zero. Moreover, we use direct integration to compute thecharacteristic curves for the solution of the Degasperis-Procesi equation wherea shockpeakon forms at a peakon-antipeakon collision. In addition to thetheoretical interest in knowing the characteristic curves, they are also usefulfor plotting multipeakon solutions, as we illustrate in several examples.

Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations