{
  "abstract": "The present work introduces a new relation between closed elastica knots with the solution of the nonlinear Schrödinger equation (NLSE) through the Hasimoto transformation. An elastica knot is defined in terms of the Frenet-Serret curvature $κ(s,t)$ as a function of the arclength $s$ along the spatial curve ${\\bf r}(s,t)$ at a fixed time $t$, which is a solution of the curvature differential equation $\\partial^{2}_{s}κ(s,t) = -\\;κ^{3}/2 + k_{0}^{4}τ_{0}^{2}\\;κ^{-3} + λ\\,k_{0}^{2}κ/2$. Here, the Frenet-Serret torsion $τ(s,t)$ satisfies the conservation law $κ^{2}(s,t)\\,τ(s,t) \\equiv k_{0}^{2}\\,τ_{0}$, while the constant of integration $λ$ is associated with the constrained variation. After briefly reviewing the Hasimoto transformation from a space curve ${\\bf r}(s,t)$ to the nonlinear Schrödinger equation (NLSE) $-\\,iD^{-1}\\partial_{t}ψ= \\partial^{2}_{s}ψ+ \\frac{1}{2}\\,|ψ|^{2}ψ$, where the constant $D$ has units of fluid circulation (m$^{2}$/sec), we show how the traveling-wave solution $ψ(s,t) = Ψ(s_{t} \\equiv s - c\\,t) \\equiv κ(s_{t})\\;\\exp[iθ(s_{t})]$ is mapped onto the curvature equation for an elastica knot, with $θ^{\\prime}(s_{t}) \\equiv c/(2D) + k_{0}^{2}τ_{0}/κ^{2}(s_{t})$ and the elastica-knot constant $k_{0}^{2}λ= -\\frac{1}{2}\\,(c/D)^{2}$ expressed in terms of the traveling-wave NLSE parameters $(c,D)$. The constraint of a closed 3D elastica knot imposes spatial periodicity conditions that introduce a unique set of knot parameters for which the NLSE traveling wave can exist. The present work shows that the NLSE traveling-wave solution on a closed elastica knot requires an extension of the classical elastica-knot parameter space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.21750",
  "author_total": 1,
  "authors": [
    {
      "name": "Alain J. Brizard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vbnwyq6syqzy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.21750",
  "fulltext_url": null,
  "id": "3vbnwyq6syqzy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.21750",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "physics.plasm-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2607.21750",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vbnwyq6syqzy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2607.21750",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.21750"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.plasm-ph",
      "subfield": "plasma"
    }
  ],
  "title": "Nonlinear Schrödinger equation on a closed 3D elastica knot",
  "title_text": "Nonlinear Schrödinger equation on a closed 3D elastica knot",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/3vbnwyq6syqzy",
  "venue": "arXiv",
  "versions": []
}