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