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
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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

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