Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

Hakkaev, Sevdzhan; Stanislavova, Milena; Stefanov, Atanas

doi:10.1111/sapm.12166

Cited by 12 publications

(23 citation statements)

References 31 publications

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“…Smooth periodic waves of the quasi-linear differential equation in (1.4) can be obtained equivalently from a semi-linear differential equation by means of the following change of coordinates [6,13,14]:…”

Section: Introductionmentioning

confidence: 99%

“…The analysis of [7] relies on the standard variational formulation of the periodic waves as critical points of energy subject to fixed momentum. The analysis of [14] relies on the coordinate transformation (1.7), which reduces the spectral stability problem of the form ∂ x Lv = λv with the self-adjoint operator L = c − U p + ∂ −2 z to the spectral problem of the form M v = λ∂ ξ v with the self-adjoint operator M = c − u p + ∂ 2 ξ . The spectral problem M v = λ∂ ξ v has been studied before in [22] (see also [15] for a generalization).…”

Section: Introductionmentioning

confidence: 99%

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Linear Instability and Uniqueness of the Peaked Periodic Wave in the Reduced Ostrovsky Equation

Geyer¹,

Pelinovsky

2019

SIAM J. Math. Anal.

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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. The main novelty of our result is 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. π −π U (z)dz = 0, Date: November 1, 2019.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Linear Instability and Uniqueness of the Peaked Periodic Wave in the Reduced Ostrovsky Equation

Geyer¹,

Pelinovsky

2019

SIAM J. Math. Anal.

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“…In these cases, travelling waves can be found in the explicit form given by the Jacobi elliptic functions after a change of coordinates [3,9]. Exploring this idea further, it was shown in [13,14,27] that the spectral stability of travelling periodic waves can be studied with the help of the eigenvalue problem M ψ = λ∂ z ψ, where M is a second-order Schrödinger operator. Independently, by using higher-order conserved quantities which exist in the integrable cases p = 1 and p = 2, it was shown in [4] that the travelling periodic waves are unconstrained minimizers of energy functions in suitable function spaces with respect to subharmonic perturbations, that is, perturbations with a multiple period to the periodic waves.…”

Section: Introductionmentioning

confidence: 99%

“…We now compare our result to the existing literature on spectral and orbital stability of periodic waves with respect to co-periodic perturbations. First, in comparison with the analysis in [14], the result of Theorem 1 is more general since p ∈ N is not restricted to the integrable cases p = 1 and p = 2. On a technical level, the method of proof of Theorem 1 is simple and robust, so that many unnecessary explicit computations from [14] are avoided.…”

Section: Introductionmentioning

confidence: 99%

Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

Geyer

Pelinovsky

2017

Lett Math Phys

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We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all smooth periodic travelling waves independent of the nonlinearity power. The argument is based on the energy convexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein–Gordon type.

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“…hence by [29,Lemma 1] it follows that L + | {φ} ⊥ ≥ 0. Since L − ≥ 0 a-priori, we conclude that δ 2 E(φ)| {φ} ⊥ ≥ 0.…”

Section: Analysis Of the Focusing Casementioning

confidence: 95%

Nondegeneracy and stability of antiperiodic bound states for fractional nonlinear Schrödinger equations

Claassen

Johnson

2019

Journal of Differential Equations

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We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schrödinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we demonstrate that the associated linearized operator is nondegenerate when restricted to antiperiodic perturbations, i.e. that its kernel is generated by the translational and gauge symmetries of the governing evolution equation. In the process, we provide a characterization of the antiperiodic ground state eigenfunctions for linear fractional Schrödinger operators on R with real-valued, periodic potentials as well as a Sturm-Liouville type oscillation theory for the higher antiperiodic eigenfunctions. variety of applications including the continuum limit of discrete models with long range interaction [39], dislocation dynamics in crystals [12], mathematical biology [44], water wave dynamics [33], and financial mathematics [14]; see also [10] for a recent discussion on applications. For such α, the nonlocal fNLS (1.1) has been introduced by Laskin [40] in the context of fractional quantum mechanics, in which one generalizes the standard Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. A rigorous derivation in the context of charge transport in bio polymers (like DNA), can be found in [39]. Finally, we point out the case α = 1 may be thought to describe the relativistic dispersion relation ω(ξ) = |ξ| 2 + m 2 , an observation recently utilized in the mathematical description of Boson-stars; see [19].Throughout our analysis, we will be concerned with solutions of the formwhere ω, c ∈ R are parameters and φ is a bounded solution to the (generally) nonlocal profile equationwhere denotes differentiation with respect to the spatial variable. When c = 0, the focusing fNLS is well-known to admit standing solitary waves that are asymptotic to zero at spatial infinity. Among such solitary wave solutions, specific attention is often paid to the positive, radially symmetric solutions typically referred to as "ground states". The stability of such ground states dates back to the work of Cazenave and Lions [13] and Weinstein [51, 52] on the classical case α = 2, where the authors use the method of concentration compactness along with the construction of appropriate Lyapunov functionals. For α ∈ (1, 2], such ground states are known to be orbitally stable provided the nonlinearity is energy sub-critical, i.e. if 0 < σ < α; see [27,52], for example. While no nontrivial localized solutions exist in the defocusing case γ = −1, it is known in the classical case α = 2 to admit so-called black solitons of the form (1.2) corresponding to monotone front-like solutions asymptotic to constants as x → ±∞. The dynamics and stability of black solitons has been studied in numerous works; see, for example, [4,23,24]. For the fractional case, the authors plan to report on the existence, nondegeneracy, and stability of black solitons in the defocusing case in a future work. The stabilit...

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Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

Cited by 12 publications

References 31 publications

Linear Instability and Uniqueness of the Peaked Periodic Wave in the Reduced Ostrovsky Equation

Linear Instability and Uniqueness of the Peaked Periodic Wave in the Reduced Ostrovsky Equation

Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

Nondegeneracy and stability of antiperiodic bound states for fractional nonlinear Schrödinger equations

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