  We continue our investigation on the asymptotic stability of the peakon. In a
first step we extend our asymptotic stability result [29] in the class of
functions whose negative part of the momentum density is supported in ] --
$\infty$, x 0 ] and the positive part in [x 0 , +$\infty$[ for some x 0 $\in$
R. In a second step this enables us to prove the asymptotic stability of
well-ordered train of antipeakons-peakons and, in particular, of the
antipeakon-peakon profile. Finally, in the appendix we prove that in the case
of a non negative momentum density the energy at the left of any given point
decays to zero as time goes to +$\infty$,. This leads to an improvement of the
asymptotic stability result stated in [29].
