{"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/dirichlet-energy-and-focusing-nls-condensates","title":"Dirichlet energy and focusing NLS condensates of minimal intensity","arxiv_id":"2412.19373","date":"2024-12-26","proceeding":null,"authors":["Marco Bertola","Alexander Tovbis"],"abstract":"We consider the family of (poly)continua $\\K$ in the upper half-plane ${\\mathbb H} $ that contain a preassigned finite {\\it anchor} set $E\\in\\mathbb H$. For a given harmonic external field we define a Dirichlet energy functional $\\mathcal I(\\mathcal K)$ and show that within each ``connectivity class'' of the family, there exists a minimizing compact $\\mathcal K^*$ consisting of critical trajectories of a quadratic differential. In many cases this quadratic differential coincides with the square of the real normalized quasimomentum differential ${\\rm d} {\\bf p}$ associated with the finite gap solutions of the focusing Nonlinear Schr\\\"{o}dinger equation (fNLS) defined by a hyperelliptic Riemann surface $\\mathfrak R$ branched at the points $E\\cup\\bar E$. The motivation for this work lies in the problem of soliton condensate of least average intensity such that a given anchor set $E$ belongs to the poly-continuum $\\mathcal K$. An fNLS soliton condensate is defined by a compact $\\mathcal K\\subset{\\mathbb H} $ (its spectral support) whereas the average intensity of the condensate is proportional to $\\mathcal I(\\mathcal K)$. We prove that the spectral support $\\mathcal K^*$ provides the fNLS soliton condensate of the least average intensity within a given ``connectivity class''.","url_abs":"https://arxiv.org/abs/2412.19373v1","url_pdf":"https://arxiv.org/pdf/2412.19373v1.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":"dirichlet-energy-and-focusing-nls-condensates","repo_url":"https://github.com/bertolamarco/NLS_soliton_condensates_minimal_intensity","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}