  It is well-known that peakons in the Camassa-Holm equation are
$H^1$-orbitally stable thanks to the presence of conserved quantities and
properties of peakons as constrained energy minimizers. By using the method of
characteristics, we prove that piecewise $C^1$ perturbations to peakons grow in
time in spite of their stability in the $H^1$-norm. We also show that the
linearized stability analysis near peakons contradicts the $H^1$-orbital
stability result, hence passage from linear to nonlinear theory is false in
$H^1$.
