Spectral instability of the peaked periodic wave in the reduced Ostrovsky equations

Geyer, Anna; Pelinovsky, Dmitry

doi:10.1090/proc/14937

Spectral instability of the peaked periodic wave in the reduced Ostrovsky equations

Authors: Anna Geyer and Dmitry E. Pelinovsky
Journal: Proc. Amer. Math. Soc. 148 (2020), 5109-5125
MSC (2010): Primary 35B35, 35Q35
DOI: https://doi.org/10.1090/proc/14937
Published electronically: September 17, 2020
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Abstract: We show that the peaked periodic traveling wave of the reduced Ostrovsky equations with quadratic and cubic nonlinearity is spectrally unstable in the space of square integrable periodic functions with zero mean and the same period. We discover that the spectrum of a linearized operator at the peaked periodic wave completely covers a closed vertical strip of the complex plane. In order to obtain this instability, we prove an abstract result on spectra of operators under compact perturbations. This justifies the truncation of the linearized operator at the peaked periodic wave to its differential part for which the spectrum is then computed explicitly.

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