Dirichlet energy and focusing NLS condensates of minimal intensity

[2412.19373] Dirichlet energy and focusing NLS condensates of minimal intensity Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2412.19373 (math) [Submitted on 26 Dec 2024 ( v1 ), last revised 11 Apr 2026 (this version, v3)] Title: Dirichlet energy and focusing NLS condensates of minimal intensity Authors: Marco Bertola , Alexander Tovbis View a PDF of the paper titled Dirichlet energy and focusing NLS condensates of minimal intensity, by Marco Bertola and 1 other authors View PDF HTML (experimental) 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 Schrödinger 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''. Comments: 37 pages, 6 (beautiful!) figures. Ver 2: 39 pages; improved introduction and summary of results, added references. Some re-rendered figures Subjects: Analysis of PDEs (math.AP) ; Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI) Cite as: arXiv:2412.19373 [math.AP] (or arXiv:2412.19373v3 [math.AP] for this version) https://doi.org/10.48550/arXiv.2412.19373 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Marco Bertola [ view email ] [v1] Thu, 26 Dec 2024 22:57:10 UTC (5,555 KB) [v2] Fri, 14 Nov 2025 21:02:37 UTC (5,639 KB) [v3] Sat, 11 Apr 2026 22:05:18 UTC (5,639 KB) Full-text links: Access Paper: View a PDF of the paper titled Dirichlet energy and focusing NLS condensates of minimal intensity, by Marco Bertola and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2024-12 Change to browse by: math math-ph math.MP nlin nlin.SI References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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