Infinitely many isolas of modulational instability for Stokes waves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915458192506880 |
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| author | Berti, Massimiliano Corsi, Livia Maspero, Alberto Ventura, Paolo |
| author_facet | Berti, Massimiliano Corsi, Livia Maspero, Alberto Ventura, Paolo |
| contents | This paper proves long-standing conjectures regarding the existence of infinitely many high-frequency modulational instability ``isolas" for a Stokes wave in arbitrary depth $ \mathtt{h} > 0 $, under longitudinal perturbations. We provide a complete characterization of the unstable spectral bands in the $L^2(\mathbb{R})$-spectrum of the water wave equations linearized around a Stokes wave of sufficiently small amplitude $ε$. The unstable spectrum is the union of isolated ``isolas" of elliptical shape, indexed by integers $ \mathtt{p}\geq 2 $, each with semiaxis of size $ |β_1^{(\mathtt{p})} (\mathtt{h})| ε^\mathtt{p}+ O(ε^{\mathtt{p}+2} )$. As first key achievement, we obtain an explicit formula for the coefficient $ β_1^{(\mathtt{p})} (\mathtt{h}) $ for any $ \mathtt{p} \geq 2 $, that remarkably depends solely on the maximal Taylor-Fourier coefficients of the Stokes wave. We provide simple expressions of the asymptotic expansion of such coefficients in the shallow-water limit $ \mathtt{h} \to 0^+ $, for any $ \mathtt{p} \geq 2 $. This allows to establish that the analytic function $β_1^{(\mathtt{p})}(\mathtt{h})$ is not zero for any $\mathtt{p} \geq 2$, by verifying that a combinatorial sum is not zero; this relies on a crucial combinatorial identity due to Koutschan, van Hoeij, and Zeilberger. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinitely many isolas of modulational instability for Stokes waves Berti, Massimiliano Corsi, Livia Maspero, Alberto Ventura, Paolo Analysis of PDEs 35B35 (Primary) 37K45, 76E30 (Secondary) This paper proves long-standing conjectures regarding the existence of infinitely many high-frequency modulational instability ``isolas" for a Stokes wave in arbitrary depth $ \mathtt{h} > 0 $, under longitudinal perturbations. We provide a complete characterization of the unstable spectral bands in the $L^2(\mathbb{R})$-spectrum of the water wave equations linearized around a Stokes wave of sufficiently small amplitude $ε$. The unstable spectrum is the union of isolated ``isolas" of elliptical shape, indexed by integers $ \mathtt{p}\geq 2 $, each with semiaxis of size $ |β_1^{(\mathtt{p})} (\mathtt{h})| ε^\mathtt{p}+ O(ε^{\mathtt{p}+2} )$. As first key achievement, we obtain an explicit formula for the coefficient $ β_1^{(\mathtt{p})} (\mathtt{h}) $ for any $ \mathtt{p} \geq 2 $, that remarkably depends solely on the maximal Taylor-Fourier coefficients of the Stokes wave. We provide simple expressions of the asymptotic expansion of such coefficients in the shallow-water limit $ \mathtt{h} \to 0^+ $, for any $ \mathtt{p} \geq 2 $. This allows to establish that the analytic function $β_1^{(\mathtt{p})}(\mathtt{h})$ is not zero for any $\mathtt{p} \geq 2$, by verifying that a combinatorial sum is not zero; this relies on a crucial combinatorial identity due to Koutschan, van Hoeij, and Zeilberger. |
| title | Infinitely many isolas of modulational instability for Stokes waves |
| topic | Analysis of PDEs 35B35 (Primary) 37K45, 76E30 (Secondary) |
| url | https://arxiv.org/abs/2405.05854 |