Growth of perturbations to the peaked periodic waves in the CamassaHolm equation,
Abstract
Peaked periodic waves in the CamassaHolm 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 CamassaHolm equation.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.09516
 Bibcode:
 2020arXiv200609516M
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 25 pages