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

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- by Anna Geyer and Dmitry E. Pelinovsky
- Proc. Amer. Math. Soc.
**148**(2020), 5109-5125 - 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.## References

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## Bibliographic 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
- ORCID: 0000-0001-5812-440X
- Email: dmpeli@math.mcmaster.ca
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5109-5125 - MSC (2010): Primary 35B35, 35Q35
- DOI: https://doi.org/10.1090/proc/14937
- MathSciNet review: 4163826