Orbital Stability of Smooth Solitary Waves for the Novikov Equation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909138316951552 |
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| author | Ehrman, Brett Johnson, Mathew A. Lafortune, Stéphane |
| author_facet | Ehrman, Brett Johnson, Mathew A. Lafortune, Stéphane |
| contents | We study the orbital stability of smooth solitary wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. These solitary waves are shown to exist as a one-parameter family (up to spatial translations) parameterized by their asymptotic endstate, and are encoded as critical points of a particular action functional. As an important step in our analysis we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operator acting on $L^2(\mathbb{R})$. We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses always holds for the Novikov equation. Together with a detailed study of the associated Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary wave solutions of the Novikov equation are nonlinearly orbitally stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10685 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbital Stability of Smooth Solitary Waves for the Novikov Equation Ehrman, Brett Johnson, Mathew A. Lafortune, Stéphane Analysis of PDEs We study the orbital stability of smooth solitary wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. These solitary waves are shown to exist as a one-parameter family (up to spatial translations) parameterized by their asymptotic endstate, and are encoded as critical points of a particular action functional. As an important step in our analysis we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operator acting on $L^2(\mathbb{R})$. We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses always holds for the Novikov equation. Together with a detailed study of the associated Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary wave solutions of the Novikov equation are nonlinearly orbitally stable. |
| title | Orbital Stability of Smooth Solitary Waves for the Novikov Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.10685 |