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E.R. Johnson and D.E. Pelinovsky

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Orbital stability of periodic waves in the class of reduced Ostrovsky equations

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Journal of Differential Equations 261 (2016), 3268-3304

**Abstract:**

Periodic travelling waves are considered in the class of reduced Ostrovsky equations that
describe low-frequency internal waves in the presence of rotation. The reduced Ostrovsky
equations with either quadratic or cubic nonlinearities can be transformed to integrable equations
of the Klein-Gordon type by means of a change of coordinates. By using the conserved
momentum and energy as well as an additional conserved quantity due to integrability, we
prove that small-amplitude periodic waves are orbitally stable with respect to subharmonic
perturbations, with period equal to an integer multiple of the period of the wave. The proof
is based on construction of a Lyapunov functional, which is convex at the periodic wave and
is conserved in the time evolution. We also show numerically that convexity of the Lyapunov
functional holds for periodic waves of arbitrary amplitudes.

**Keywords**:

Reduced Ostrovsky equation; short-pulse equation; periodic waves; orbital stability;
conserved quantities; Floquet-Bloch spectrum.