Modulational Instability of Small Amplitude Periodic Traveling Waves in the Novikov Equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915426199404544 |
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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 spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the $L^2(\mathbb{R})$-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number if greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13969 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modulational Instability of Small Amplitude Periodic Traveling Waves in the Novikov Equation Ehrman, Brett Johnson, Mathew A. Lafortune, Stéphane Analysis of PDEs Exactly Solvable and Integrable Systems 35Q51 35Q51 35Q51, 35B10, 35B35, 35Q35 We study the spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the $L^2(\mathbb{R})$-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number if greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization. |
| title | Modulational Instability of Small Amplitude Periodic Traveling Waves in the Novikov Equation |
| topic | Analysis of PDEs Exactly Solvable and Integrable Systems 35Q51 35Q51 35Q51, 35B10, 35B35, 35Q35 |
| url | https://arxiv.org/abs/2409.13969 |