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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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Additional Information

Anna Geyer
Affiliation: Delft Institute of Applied Mathematics, Faculty Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands
MR Author ID: 970769
Email: A.Geyer@tudelft.nl

Dmitry E. Pelinovsky
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1; and Department of Applied Mathematics, Nizhny Novgorod State Technical University, 24 Minin Street, 603950 Nizhny Novgorod, Russia
MR Author ID: 355614
Email: dmpeli@math.mcmaster.ca

DOI: https://doi.org/10.1090/proc/14937
Keywords: Peaked periodic wave, reduced Ostrovsky equation, spectral instability
Received by editor(s): July 18, 2019
Received by editor(s) in revised form: July 31, 2019, and October 23, 2019
Published electronically: September 17, 2020
Additional Notes: The second author acknowledges financial support from the state task program in the sphere of scientific activity of the Ministry of Science and Higher Education of the Russian Federation (Task No. FSWE-2020-0007) and from the grant of the president of the Russian Federation for the leading scientific schools (grant No. NSH-2485.2020.5).
Communicated by: Catherine Sulem
Article copyright: © Copyright 2020 American Mathematical Society