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.