Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

Geyer, Anna; Pelinovsky, Dmitry E.

doi:10.1007/s11005-017-0941-3

Cited by 25 publications

(41 citation statements)

References 28 publications

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“…Most of the content in Section 4.1 is contained in some recent papers, but we summarize it for the readers' convenience. Note that our reasoning applies equally well to both the fourth-and second-order versions of the eigenvalue problems, see (15) and (16).…”

Section: Stabilit Y Of the Wavessupporting

confidence: 59%

“…A full description of its periodic waves as well their stability can be found in the recent papers. 15,16 For the full Ostrovsky model under consideration, Liu and Ohta, 17 and by a slightly different method, Liu 18 have established the orbital stability for the classical Ostrovsky's equation (i.e., = 2) for large speeds. Another set of stability result, sometimes referred to as weak orbital stability, is given in Ref.…”

Section: Stability Resultsmentioning

confidence: 99%

“…Clearly, ( + , ( + )) = 4 (ℝ) is unbounded, but self-adjoint operator on 2 (ℝ). Spectral instability here is understood as the existence of a non-trivial pair ( , ) ∶ ℜ > 0, ≠ 0, ∈ ( + ), so that (15) is satisfied. Spectral stability means nonexistence of such pair.…”

Section: Stability Resultsmentioning

confidence: 99%

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On the ground states of the Ostrovskyi equation and their stability

Posukhovskyi

Stefanov

2020

Stud Appl Math

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The Ostrovskyi (Ostrovskyi-Vakhnenko/short pulse) equations are ubiquitous models in mathematical physics. They describe water waves under the action of a Coriolis force as well as the amplitude of a "short" pulse in an optical fiber. In this paper, we rigorously construct ground traveling waves for these models as minimizers of the Hamiltonian functional for any fixed 2 norm. The existence argument proceeds via the method of compensated compactness, but it requires surprisingly detailed Fourier analysis arguments to rule out the nonvanishing of the limits of the minimizing sequences. We show that all of these waves are weakly nondegenerate and spectrally stable. K E Y W O R D S ground states, Ostrovskyi equation, stability 2 0 1 0 M A

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Section: Stabilit Y Of the Wavessupporting

confidence: 59%

Section: Stability Resultsmentioning

confidence: 99%

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On the ground states of the Ostrovskyi equation and their stability

Posukhovskyi

Stefanov

2020

Stud Appl Math

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“…Spectral stability of smooth periodic waves with respect to perturbations of the same period was proven both for (1.1) and (1.2) in [7,14]. 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 .…”

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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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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“…The works [5,15] and [21] both exploit the integrable structure, so our result could be considered an alternate proof which does not uses integrability, but instead relies mainly on an invariant subspace decomposition and an elementary Krein-signature-type argument. See also the recent work [16] for related arguments.…”

mentioning

confidence: 96%

Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations

Gustafson

Coz

Tsai

2017

Applied Mathematics Research eXpress

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We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schrödinger equations. First, we give global variational characterizations of each of these periodic waves, which in particular provide alternate proofs of their orbital stability with respect to same-period perturbations, restricted to certain subspaces. Second, we prove the spectral stability of the cnoidal waves (in a certain parameter range) and snoidal waves against same-period perturbations, thus providing an alternate proof of this (known) fact, which does not rely on complete integrability. Third, we give a rigorous version of a formal asymptotic calculation of Rowlands to establish the instability of a class of real-valued periodic waves in 1D, which includes the cnoidal waves of the 1D cubic focusing nonlinear Schrödinger equation, against perturbations with period a large multiple of their fundamental period. Finally, we develop a numerical method to compute the minimizers of the energy with fixed mass and momentum constraints. Numerical experiments support and complete our analytical results.

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Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

Cited by 25 publications

References 28 publications

On the ground states of the Ostrovskyi equation and their stability

On the ground states of the Ostrovskyi equation and their stability

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

Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations

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