First isola of modulational instability of Stokes waves in deep water
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910308175446016 |
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| author | Berti, Massimiliano Maspero, Alberto Ventura, Paolo |
| author_facet | Berti, Massimiliano Maspero, Alberto Ventura, Paolo |
| contents | We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from $\mathtt{i}\frac34$. Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory we reduce to search the eigenvalues of a $2\times 2$ Hamiltonian and reversible matrix which has eigenvalues with non-zero real part if and only if a certain analytic function is not identically zero. In deep water we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_14689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First isola of modulational instability of Stokes waves in deep water Berti, Massimiliano Maspero, Alberto Ventura, Paolo Analysis of PDEs 76B15 (Primary), 35B35, 76E99 (Secondary) We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from $\mathtt{i}\frac34$. Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory we reduce to search the eigenvalues of a $2\times 2$ Hamiltonian and reversible matrix which has eigenvalues with non-zero real part if and only if a certain analytic function is not identically zero. In deep water we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one. |
| title | First isola of modulational instability of Stokes waves in deep water |
| topic | Analysis of PDEs 76B15 (Primary), 35B35, 76E99 (Secondary) |
| url | https://arxiv.org/abs/2401.14689 |