A. Madiyeva and D.E. Pelinovsky

Growth of perturbations to the peaked periodic waves in the Camassa-Holm equation

SIAM Journal of Mathematical Analysis 53 (2021), 3016-3039

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 H¹ and W^1,∞.norms. Dynamics of perturbations in H¹. 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,∞ norm and may blow up in a finite time in the nonlinear evolution of the Camassa-Holm equation.

Keywords:
Camassa-Holm equation; peaked periodic waves; orbital stability; wave breaking;