{
  "abstract": "We study polynomial Hamiltonian systems whose Hamiltonians consist of two weighted products of canonical modes minus a polynomial in their powers. This class extends the structure of familiar genus-one coupled-mode systems from nonlinear optics. We construct local analytic solutions in terms of sigma functions and Abelian integrals on smooth odd-degree hyperelliptic curves, subject to stated regularity conditions. We associate Onishi's determinant identities with the conservation-law structure of these systems, extending the connection previously identified between Frobenius-Stickelberger determinants and conserved quantities in genus-one nonlinear optics. Explicit genus-two and genus-three examples include bounded oscillatory scalar motion. We also classify the Hamiltonian polynomials that eliminate the additional mode exponentials and show that their elementary modes solve second-order equations with self-generating polynomial potentials. We evaluate the solutions using Generapy, a purpose-built Python package for algebraic curves and Abelian functions, and validate them against independent fourth-order Runge-Kutta integration.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22784",
  "author_total": 1,
  "authors": [
    {
      "name": "Graham Hesketh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9snq8wapsnref.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22784",
  "fulltext_url": null,
  "id": "9snq8wapsnref",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22784",
  "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",
      "nlin.SI"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7222951129",
  "pdf_url": "https://arxiv.org/pdf/2610.22784",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9snq8wapsnref.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2610.22784",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22784"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Hyperelliptic sigma solutions of polynomial Hamiltonian systems",
  "title_text": "Hyperelliptic sigma solutions of polynomial Hamiltonian systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:15Z",
  "url": "https://research.draconitic.com/paper/9snq8wapsnref",
  "venue": "arXiv",
  "versions": []
}