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

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

## 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

- G. Bruell and R. N. Dhara,
*Waves of maximal height for a class of nonlocal equations with homogeneous symbol*, arXiv:1810.00248v1, 2018. - Theo Bühler and Dietmar A. Salamon,
*Functional analysis*, Graduate Studies in Mathematics, vol. 191, American Mathematical Society, Providence, RI, 2018. MR**3823238**, DOI 10.1090/gsm/191 - Carmen Chicone and Yuri Latushkin,
*Evolution semigroups in dynamical systems and differential equations*, Mathematical Surveys and Monographs, vol. 70, American Mathematical Society, Providence, RI, 1999. MR**1707332**, DOI 10.1090/surv/070 - Mats Ehrnström, Mathew A. Johnson, and Kyle M. Claassen,
*Existence of a highest wave in a fully dispersive two-way shallow water model*, Arch. Ration. Mech. Anal.**231**(2019), no. 3, 1635–1673. MR**3902471**, DOI 10.1007/s00205-018-1306-5 - E. R. Johnson and R. H. J. Grimshaw,
*The modified reduced Ostrovsky equation: integrability and breaking*, Phys. Rev. E**88**(2014), 021201(R) (5 pages). - Edward R. Johnson and Dmitry E. Pelinovsky,
*Orbital stability of periodic waves in the class of reduced Ostrovsky equations*, J. Differential Equations**261**(2016), no. 6, 3268–3304. MR**3527630**, DOI 10.1016/j.jde.2016.05.026 - Anna Geyer and Dmitry E. Pelinovsky,
*Spectral stability of periodic waves in the generalized reduced Ostrovsky equation*, Lett. Math. Phys.**107**(2017), no. 7, 1293–1314. MR**3685174**, DOI 10.1007/s11005-017-0941-3 - Anna Geyer and D. Pelinovsky,
*Linear instability and uniqueness of the peaked periodic wave in the reduced Ostrovsky equation*, SIAM J. Math. Anal.**51**(2019), no. 2, 1188–1208. MR**3936896**, DOI 10.1137/18M117978X - R. Grimshaw,
*Evolution equations for weakly nonlinear, long internal waves in a rotating fluid*, Stud. Appl. Math.**73**(1985), no. 1, 1–33. MR**797556**, DOI 10.1002/sapm19857311 - R. H. J. Grimshaw, Karl Helfrich, and E. R. Johnson,
*The reduced Ostrovsky equation: integrability and breaking*, Stud. Appl. Math.**129**(2012), no. 4, 414–436. MR**2993126**, DOI 10.1111/j.1467-9590.2012.00560.x - R. H. J. Grimshaw, L. A. Ostrovsky, V. I. Shrira, and Yu. A. Stepanyants,
*Long nonlinear surface and internal gravity waves in a rotating ocean*, Surv. Geophys.**19**(1998), 289–338. - Roger Grimshaw and Dmitry Pelinovsky,
*Global existence of small-norm solutions in the reduced Ostrovsky equation*, Discrete Contin. Dyn. Syst.**34**(2014), no. 2, 557–566. MR**3094592**, DOI 10.3934/dcds.2014.34.557 - Sevdzhan Hakkaev, Milena Stanislavova, and Atanas Stefanov,
*Periodic traveling waves of the regularized short pulse and Ostrovsky equations: existence and stability*, SIAM J. Math. Anal.**49**(2017), no. 1, 674–698. MR**3614681**, DOI 10.1137/15M1037901 - S. Hakkaev, M. Stanislavova, and A. Stefanov,
*Spectral stability for classical periodic waves of the Ostrovsky and short pulse models*, Stud. Appl. Math.**139**(2017), no. 3, 405–433. MR**3708844**, DOI 10.1111/sapm.12166 - Aleksey Kostenko and Noema Nicolussi,
*On the Hamiltonian-Krein index for a non-self-adjoint spectral problem*, Proc. Amer. Math. Soc.**146**(2018), no. 9, 3907–3921. MR**3825844**, DOI 10.1090/proc/14048 - Yue Liu, Dmitry Pelinovsky, and Anton Sakovich,
*Wave breaking in the Ostrovsky-Hunter equation*, SIAM J. Math. Anal.**42**(2010), no. 5, 1967–1985. MR**2684307**, DOI 10.1137/09075799X - S. P. Nikitenkova, Yu. A. Stepanyants, and L. M. Chikhladze,
*Solutions of a modified Ostrovskiĭ equation with a cubic nonlinearity*, Prikl. Mat. Mekh.**64**(2000), no. 2, 276–284 (Russian, with Russian summary); English transl., J. Appl. Math. Mech.**64**(2000), no. 2, 267–274. MR**1773718**, DOI 10.1016/S0021-8928(00)00048-4 - L. A. Ostrovsky,
*Nonlinear internal waves in a rotating ocean*, Okeanologia**18**(1978), 181–191. - L. Ostrovsky, E. Pelinovsky, V. Shrira, and Y. Stepanyants,
*Beyond the KdV: post-explosion development*, Chaos**25**(2015), no. 9, 097620, 13. MR**3396218**, DOI 10.1063/1.4927448 - Roman Shvidkoy and Yuri Latushkin,
*The essential spectrum of the linearized 2D Euler operator is a vertical band*, Advances in differential equations and mathematical physics (Birmingham, AL, 2002) Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 299–304. MR**1991549**, DOI 10.1090/conm/327/05822 - Atanas Stefanov, Yannan Shen, and P. G. Kevrekidis,
*Well-posedness and small data scattering for the generalized Ostrovsky equation*, J. Differential Equations**249**(2010), no. 10, 2600–2617. MR**2718712**, DOI 10.1016/j.jde.2010.05.015 - Milena Stanislavova and Atanas Stefanov,
*On the spectral problem $\mathcal Lu=\lambda u’$ and applications*, Comm. Math. Phys.**343**(2016), no. 2, 361–391. MR**3477342**, DOI 10.1007/s00220-015-2542-2

## 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
- 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