Linear instability and uniqueness of the peaked periodic wave in the reduced Ostrovsky equation
Anna Geyer (TU Delft - Mathematical Physics)
Dmitry Pelinovsky (McMaster University, Nizhni Novgorod State Technical University)
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Abstract
The stability of the peaked periodic wave 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 L
2
norm of co-periodic perturbations to the peaked periodic wave, from which it follows that the peaked periodic wave is linearly unstable. We also prove that the peaked periodic wave with a parabolic profile is the unique peaked wave in the space of periodic L
2
functions with zero mean and a single minimum per period.