Orbital and asymptotic stability of a train of peakons for the Novikov equation

Palacios, José M.

doi:10.3934/dcds.2020372

Cited by 10 publications

(9 citation statements)

References 30 publications

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“…[8] on 𝑊 1,∞ and the orbital stability results of Refs. [7,29,30] on 𝐻 1 (ℝ). Per the discussion in the Introduction section, there is no easy relation between linear and orbital stability for peakon solutions.…”

Section: Discussionmentioning

confidence: 99%

“…The unicity is a consequence of the unicity of the solution of the initial value problem (30). By the chain rule, we obtain…”

Section: Spectral and Linear Instability On 𝑳 𝟐 (ℝ)mentioning

confidence: 97%

“…Note that 𝑣(0, 𝑡) = 0 since 𝔳 0 (0) = 𝑞(0, 𝑡) = 0. The unicity result comes from the fact that the solution (31) is the unique solution to Equation (30).…”

Section: Spectral and Linear Stability On 𝑯 𝟏 (ℝ)mentioning

confidence: 99%

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Spectral and linear stability of peakons in the Novikov equation

Lafortune

2024

Stud Appl Math

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The Novikov equation is a peakon equation with cubic nonlinearity, which, like the Camassa–Holm and the Degasperis–Procesi, is completely integrable. In this paper, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in . To do so, we start with a linearized operator defined on and extend it to a linearized operator defined on weaker functions in . The spectrum of the linearized operator in is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on and linearly and spectrally stable on . The result on is in agreement with previous work about linear instability and our result on is in line with past work on orbital stability.

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Section: Discussionmentioning

confidence: 99%

“…The unicity is a consequence of the unicity of the solution of the initial value problem (30). By the chain rule, we obtain…”

Section: Spectral and Linear Instability On 𝑳 𝟐 (ℝ)mentioning

confidence: 97%

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Spectral and linear stability of peakons in the Novikov equation

Lafortune

2024

Stud Appl Math

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“…The literature surrounding the Novikov equation is not yet quite as overwhelming as for the CH and DP equations, but there is no shortage of articles about PDE-analytic questions, in addition to the one just mentioned [253,290,305,176,147,314,306,135,150,195,315,196,307,53,304,143,34,149,328,302,63,228,278,207]. Stability of peakons has been considered by several researchers [215,299,257,258,54,55], and likewise solitons [237,206,260,259,301,229,326] and integrability aspects [281,30,186,29,273]. However, the numerical analysis community has not yet jumped on the bandwagon; we are only aware of two (rather similar) studies [57,58].…”

Section: The Novikov Equationmentioning

confidence: 99%

A view of the peakon world through the lens of approximation theory

Lundmark,

Szmigielski

2022

Preprint

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Peakons (peaked solitons) are a particular type of solution admitted by certain nonlinear PDEs, most famously the Camassa-Holm shallow water wave equation. These solutions take the form of a train of peak-shaped waves, interacting in a particle-like fashion. In this article we give an overview of the mathematics of peakons, with special emphasis on the connections to classical problems in analysis, such as Padé approximation, mixed Hermite-Padé approximation, multi-point Padé approximation, continued fractions of Stieltjes type and (bi)orthogonal polynomials. The exposition follows the chronological development of our understanding, exploring the peakon solutions of the Camassa-Holm, Degasperis-Procesi, Novikov, Geng-Xue and modified Camassa-Holm (FORQ) equations. All of these paradigm examples are integrable systems arising as the compatibility condition of a Lax pair, and a recurring theme in the context of peakons is the need to properly interpret these Lax pairs in the sense of Schwartz's theory of distributions. We trace out the path leading from distributional Lax pairs to explicit formulas for peakon solutions via a multitude of approximation-theoretic problems, and illustrate the dynamics of peakons with graphics. ContentsThe modified Camassa-Holm equation and distributional Lax integrability 31

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“…Nevertheless, it is easy to check that it can actually be sharpened to depend only on u0,x L ∞ . Here, since u0 ∈ Y+ we have u0,xL ∞ ≤ u0 L ∞ ≤ u0 H 1 .The interested reader can consult to[29] for a simplification of this proof without the sign assumption of the momentum density.…”

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confidence: 99%

Asymptotic stability of peakons for the Novikov equation

Palacios¹

2020

Journal of Differential Equations

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Orbital and asymptotic stability of a train of peakons for the Novikov equation

Cited by 10 publications

References 30 publications

Spectral and linear stability of peakons in the Novikov equation

Spectral and linear stability of peakons in the Novikov equation

A view of the peakon world through the lens of approximation theory

Asymptotic stability of peakons for the Novikov equation

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