  A wide class of nonlinear dispersive wave equations are shown to possess a
novel type of peakon solution in which the amplitude and speed of the peakon
are time-dependent. These novel dynamical peakons exhibit a wide variety of
different behaviours for their amplitude, speed, and acceleration, including an
oscillatory amplitude and constant speed which describes a peakon breather.
Examples are presented of families of nonlinear dispersive wave equations that
illust rate various interesting behaviours, such as asymptotic travelling-wave
peakons, dissipating/anti-dissipating peakons, direction-reversing peakons,
runaway and blow up peakons, among others.
