F. Natali and D.E. Pelinovsky

Instability of H1-stable peakons in the Camassa-Holm equation

It is well-known that peakons in the Camassa–Holm equation are H1-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 C1 perturbations to peakons grow in time in spite of their stability in the H1-norm. We also show that the linearized stability analysis near peakons contradicts the H1-orbital stability result, hence passage from linear to nonlinear theory is false in H1.

Camassa-Holm equation; peakons; orbital stability and instability.