Spectral and linear stability of peakons in the Novikov equation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910400671383552 |
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| author | Lafortune, Stéphane |
| author_facet | Lafortune, Stéphane |
| contents | The Novikov equation is a peakon equation with cubic nonlinearity which, like the Camassa-Holm and the Degasperis-Procesi, is completely integrable. In this article, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in $L^2(\mathbb{R})$. To do so, we start with a linearized operator defined on $H^1(\mathbb{R})$ and extend it to a linearized operator defined on weaker functions in $L^2(\mathbb{R})$. The spectrum of the linearized operator in $L^2(\mathbb{R})$ is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on $W^{1,\infty}(\mathbb{R})$ and linearly and spectrally stable on $H^1(\mathbb{R})$. The result on $W^{1,\infty}(\mathbb{R})$ are in agreement with previous work about linear stability, while our results on $H^1(\mathbb{R})$ are in agreement with the orbital stability obtained previously. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06655 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral and linear stability of peakons in the Novikov equation Lafortune, Stéphane Analysis of PDEs Exactly Solvable and Integrable Systems 35B35 (Primary) 35C08, 35Q35 (Secondary) The Novikov equation is a peakon equation with cubic nonlinearity which, like the Camassa-Holm and the Degasperis-Procesi, is completely integrable. In this article, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in $L^2(\mathbb{R})$. To do so, we start with a linearized operator defined on $H^1(\mathbb{R})$ and extend it to a linearized operator defined on weaker functions in $L^2(\mathbb{R})$. The spectrum of the linearized operator in $L^2(\mathbb{R})$ is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on $W^{1,\infty}(\mathbb{R})$ and linearly and spectrally stable on $H^1(\mathbb{R})$. The result on $W^{1,\infty}(\mathbb{R})$ are in agreement with previous work about linear stability, while our results on $H^1(\mathbb{R})$ are in agreement with the orbital stability obtained previously. |
| title | Spectral and linear stability of peakons in the Novikov equation |
| topic | Analysis of PDEs Exactly Solvable and Integrable Systems 35B35 (Primary) 35C08, 35Q35 (Secondary) |
| url | https://arxiv.org/abs/2308.06655 |