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 W1,∞.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 W1,∞ 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;