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

Geyer, Anna; Pelinovsky, Dmitry E.

doi:10.1090/proc/14937

Cited by 16 publications

(8 citation statements)

References 21 publications

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“…(31) shows that with enough regularity, the two bands in Theorem 2.1 can be made as close as one wants to each other. A spectrum consisting of a strip about the imaginary axis also occurs in the study of the peaked periodic wave of both versions of the reduced Ostrovsky equations [33]. Although our solutions are not periodic, the nature of the result is similar.…”

Section: Resultssupporting

confidence: 53%

The Stability of the $b$-family of Peakon Equations

Charalampidis¹,

Parker²,

Kevrekidis³

et al. 2020

Preprint

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In the present work we revisit the b-family model of peakon equations, containing as special cases the b = 2 (Camassa-Holm) and b = 3 (Degasperis-Procesi) integrable examples. We establish information about the point spectrum of the peakon solutions and notably find that for suitably smooth perturbations there exists point spectrum in the right half plane rendering the peakons unstable for b < 1. We explore numerically these ideas in the realm of fixed-point iterations, spectral stability analysis and time-stepping of the model for the different parameter regimes. In particular, we identify exact, stationary (spectrally stable) lefton solutions for b < −1, and for −1 < b < 1, we dynamically identify ramp-cliff solutions as dominant states in this regime. We complement our analysis by examining the breakup of smooth initial data into stable peakons for b > 1. While many of the above dynamical features had been explored in earlier studies, in the present work, we supplement them, wherever possible, with spectral stability computations.

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Section: Resultssupporting

confidence: 53%

The Stability of the $b$-family of Peakon Equations

Charalampidis¹,

Parker²,

Kevrekidis³

et al. 2020

Preprint

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show abstract

“…This type of dispersive terms tends to behave smoothly at far field and worse at the origin. Solutions to the type of equation (1.1) with negative orders feature wave breaking and peaked periodic waves; see discussions in [13,15]. It should be mentioned that for a linear operator of order strictly smaller than −1, the regularity of the highest, periodic, traveling-wave solution does not depend on the order operator; see Theorem 4.5.…”

Section: Introductionmentioning

confidence: 98%

Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

2020

Preprint

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“…Linear and nonlinear instability of peaked periodic waves with respect to peaked periodic perturbations was shown for the CH equation in [35]. Spectral and linear instability of peaked periodic waves for the reduced Ostrovsky equation was proven in [15,16]. Instability of peaked periodic waves in the DP equation or in the generalized CH equation is still open for further studies.…”

Section: Introductionmentioning

confidence: 99%

Stability of smooth periodic traveling waves in the Degasperis-Procesi equation

Geyer¹,

Pelinovsky²

2022

Preprint

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We derive a precise energy stability criterion for smooth periodic waves in the Degasperis-Procesi (DP) equation. Compared to the Camassa-Holm (CH) equation, the number of negative eigenvalues of an associated Hessian operator changes in the existence region of smooth perodic waves. We utilize properties of the period function with respect to two parameters in order to obtain a smooth existence curve for the family of smooth periodic waves of a fixed period. The energy stability condition is derived on parts of this existence curve which correspond to either one or two negative eigenvalues of the Hessian operator. We show numerically that the energy stability condition is satisfied on either part of the curve and prove analytically that it holds in a neighborhood of the boundary of the existence region of smooth periodic waves.

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

Cited by 16 publications

References 21 publications

The Stability of the $b$-family of Peakon Equations

The Stability of the $b$-family of Peakon Equations

Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

Stability of smooth periodic traveling waves in the Degasperis-Procesi equation

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