{
  "abstract": "We study the polynomial generalized Korteweg--de Vries family \\[ u_t+P_m(u)u_x+κu_{xxx}=0,\\qquad κ\\neq0, \\] where $P_m$ is a real polynomial of degree $m\\ge1$. The purpose is not to generate isolated closed-form waves, but to identify the structural threshold at which pole-type Painlevé behavior and elliptic traveling-wave geometry are simultaneously lost. A Weiss--Tabor--Carnevale dominant-balance calculation gives the universal principal exponent $p=-2/m$. Hence only $m=1$ and $m=2$ can possess a principal Laurent branch with integer pole order; their resonance sets are respectively $\\{-1,4,6\\}$ and $\\{-1,3,4\\}$. These two sectors are, after affine and Galilean transformations, the KdV and modified KdV equations, so their compatibility conditions and complete-integrability structures are inherited from the classical hierarchies. In contrast, every degree $m\\ge3$ has a fractional leading exponent and therefore fails the strong WTC pole criterion at the first step. The results provide a concise bridge between singularity analysis, Hamiltonian form, algebraic-curve genus and exact nonlinear waves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: SIGMA: SYMMETRY, 2026 »»»»» INTEGRABILITY and GEOMETRY: METHODS and APPLICATIONS",
  "arxiv": "2609.12006",
  "author_total": 1,
  "authors": [
    {
      "name": "Alvaro H. Salas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eg0vx87pw7qxa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12006",
  "fulltext_url": null,
  "id": "eg0vx87pw7qxa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12006",
  "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",
      "nlin.SI"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7212840100",
  "pdf_url": "https://arxiv.org/pdf/2609.12006",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eg0vx87pw7qxa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:07Z",
      "native_id": "2609.12006",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12006"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Painlevé Integrability, Hamiltonian Structure and Exact Elliptic Reductions of a Generalized Nonlinear Wave Equation",
  "title_text": "Painlevé Integrability, Hamiltonian Structure and Exact Elliptic Reductions of a Generalized Nonlinear Wave Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/eg0vx87pw7qxa",
  "venue": "arXiv",
  "versions": []
}