{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/infinitely-many-isolas-of-modulational","title":"Infinitely many isolas of modulational instability for Stokes waves","arxiv_id":"2405.05854","date":"2024-05-09","proceeding":null,"authors":["Massimiliano Berti","Livia Corsi","Alberto Maspero","Paolo Ventura"],"abstract":"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 $\\epsilon$. The unstable spectrum is the union of isolated ``isolas\" of elliptical shape, indexed by integers $ \\mathtt{p}\\geq 2 $, each with semiaxis of size $ |\\beta_1^{(\\mathtt{p})} (\\mathtt{h})| \\epsilon^\\mathtt{p}+ O(\\epsilon^{\\mathtt{p}+2} )$. As first key achievement, we obtain an explicit formula for the coefficient $ \\beta_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 $\\beta_1^{(\\mathtt{p})}(\\mathtt{h})$ 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.","url_abs":"https://arxiv.org/abs/2405.05854v2","url_pdf":"https://arxiv.org/pdf/2405.05854v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"infinitely-many-isolas-of-modulational","repo_url":"https://git-scm.sissa.it/amaspero/isolas","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}