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A. Geyer and D.E. Pelinovsky

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

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**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 L^{2} 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 L^{2} functions with zero mean and a single minimum per period.

**Keywords**:

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