{
  "abstract": "We establish the complete integrability of the defocusing Calogero--Sutherland derivative nonlinear Schrödinger equation \\begin{equation}\\label{CS-abstract}\\tag{CS} i\\partial_t u +\\partial_x^2 u + 2i u \\partial_x\\,Π\\!\\left(|u|^2\\right)=0,\\qquad x\\in\\mathbb T, \\end{equation} by constructing a nonlinear Fourier transform \\begin{equation*} Φ: u\\in L^2_+(\\mathbb T)\\mapsto\\ (ζ_n(u))_{n\\geq 0}\\in \\ell^2(\\mathbb N_0),\\qquad ζ_n(u)=\\sqrt{γ_n(u)}\\;e^{\\,i\\arg\\langle u\\mid f_n\\rangle}, \\end{equation*} built from the eigenvalues $(λ_n)$, the spectral gaps $γ_n:=λ_n-λ_{n-1}-1\\geq 0$ and the eigenfunctions $(f_n)$ of the Lax operator $L_u$. We prove that $Φ$ is a norm-preserving homeomorphism. In these coordinates, the flow becomes the explicit rotation $ζ_n(t)=e^{-iω_n t}ζ_n(0)$ for all $n$, whose frequencies $ω_n$ depend only on the conserved spectrum $(λ_n)$. As consequences, every orbit of \\eqref{CS-abstract} is precompact in $L^2_+(\\mathbb T)$, and $t\\mapsto u(t)$ is Bohr almost periodic. For finite-gap data, we obtain quasi-periodicity in time and a sharp criterion for genuine periodicity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23625",
  "author_total": 1,
  "authors": [
    {
      "name": "Rana Badreddine"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bd1bd8dg2wqc2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23625",
  "fulltext_url": null,
  "id": "bd1bd8dg2wqc2",
  "issns": [],
  "landing_url": "https://hal.science/hal-05776004v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-ap"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7214005562",
  "pdf_url": "https://hal.science/hal-05776004/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd1bd8dg2wqc2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:46Z",
      "native_id": "hal-05776004",
      "source": "hal",
      "url": "https://hal.science/hal-05776004v1"
    },
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23625",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23625"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Complete integrability of the Calogero--Sutherland DNLS equation on $L^2_+(\\mathbb T)$",
  "title_text": "Complete integrability of the Calogero--Sutherland DNLS equation on L^2_+(𝕋)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/bd1bd8dg2wqc2",
  "venue": "HAL",
  "versions": []
}