Instability of H 1-stable peakons in the Camassa–Holm equation

It is well-known that peakons in the Camassa–Holm equation are H 1 -orbitally stable thanks to 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 .