A. Geyer and D.E. Pelinovsky

Linear and nonlinear instability of the peaked periodic wave in the reduced Ostrovsky equation


Abstract:
Stability of the peaked periodic waves in the reduced Ostrovsky equation has remained an open problem for a long time. In order to solve this problem we obtain sharp bounds on the exponential growth of the L2 norm of co-periodic perturbations to the peaked periodic wave, from which it follows that the peaked periodic wave is orbitally unstable. We also prove that the peaked periodic wave with parabolic profile is the unique peaked wave in the space of periodic L2 functions with zero mean and a single minimum per period.

Keywords:
Reduced Ostrovsky equation; peaked periodic waves; linear instability; nonlinear instability;