Growth of Perturbations to the Peaked Periodic Waves in the Camassa-Holm Equation

Abstract

Peaked periodic waves in the Camassa-Holm equation are revisited. Linearized evolution equations are derived for perturbations to the peaked periodic waves and linearized instability is proven both in $H^1$ and $W^{1,\\infty}$ norms. Dynamics of perturbations in $H^1$ is related to the existence of two conserved quantities and is bounded in the full nonlinear system due to these conserved quantities. On the other hand, perturbations to the peaked periodic wave grow in $W^{1,\\infty}$ norm and may blow up in a finite time in the nonlinear evolution of the Camassa-Holm equation.