Orbital stability of smooth solitary waves for the modified Camassa-Holm equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908801193476096 |
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| author | Deng, Xijun Lafortune, Stéphane Liu, Zhisu |
| author_facet | Deng, Xijun Lafortune, Stéphane Liu, Zhisu |
| contents | In this paper, we explore the orbital stability of smooth solitary wave solutions to the modified Camassa-Holm equation with cubic nonlinearity. These solutions, which exist on a nonzero constant background $k$, are unique up to translation for each permissible value of $k$ and wave speed. By leveraging the Hamiltonian nature of the modified Camassa-Holm equation and employing three conserved functionals-comprising an energy and two Casimirs, we establish orbital stability through an analysis of the Vakhitov-Kolokolov condition. This stability pertains to perturbations of the momentum variable in $H^1(\mathbb{R})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19342 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbital stability of smooth solitary waves for the modified Camassa-Holm equation Deng, Xijun Lafortune, Stéphane Liu, Zhisu Analysis of PDEs Exactly Solvable and Integrable Systems 35B35, 35C08, 35Q51, 35Q35, 47A10 In this paper, we explore the orbital stability of smooth solitary wave solutions to the modified Camassa-Holm equation with cubic nonlinearity. These solutions, which exist on a nonzero constant background $k$, are unique up to translation for each permissible value of $k$ and wave speed. By leveraging the Hamiltonian nature of the modified Camassa-Holm equation and employing three conserved functionals-comprising an energy and two Casimirs, we establish orbital stability through an analysis of the Vakhitov-Kolokolov condition. This stability pertains to perturbations of the momentum variable in $H^1(\mathbb{R})$. |
| title | Orbital stability of smooth solitary waves for the modified Camassa-Holm equation |
| topic | Analysis of PDEs Exactly Solvable and Integrable Systems 35B35, 35C08, 35Q51, 35Q35, 47A10 |
| url | https://arxiv.org/abs/2412.19342 |