{
  "abstract": "We relate a complex Hamiltonian extension of the asymmetric conservative coherent coupler to three-mode degenerate four-wave mixing. A trajectory-dependent reciprocal gauge preserves the modal products and gives an autonomous normal form with meromorphic envelopes; we construct a projection from degenerate four-wave mixing onto this normal form. Building on established elliptic power-transfer and phase descriptions, we reconstruct the complex envelopes explicitly in Weierstrass $σ$ and $ζ$ functions for unequal propagation constants and self-phase coefficients with symmetric cross-phase coupling. This representation exposes the noninteger powers of sigma ratios introduced generically by unequal self-phase coefficients, and their removal by the gauge. The lifted modes are exponentially gauged Kronecker kernels, equivalently genus-one Baker-Akhiezer functions, and the conserved Hamiltonian product relation is represented by a four-point Frobenius-Stickelberger determinant identity. These mode-level relationships place the coupler within a wider family of integrable continuous-wave systems; numerical comparisons verify the formulae and their evaluation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.19216",
  "author_total": 1,
  "authors": [
    {
      "name": "Graham Hesketh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bx0sghmwxwhsr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.19216",
  "fulltext_url": null,
  "id": "bx0sghmwxwhsr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.19216",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7161774521",
  "pdf_url": "https://arxiv.org/pdf/2605.19216",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bx0sghmwxwhsr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2605.19216",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.19216"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Elliptic solutions, gauge normal forms and a degenerate four-wave-mixing projection for coherent couplers",
  "title_text": "Elliptic solutions, gauge normal forms and a degenerate four-wave-mixing projection for coherent couplers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:35Z",
  "url": "https://research.draconitic.com/paper/bx0sghmwxwhsr",
  "venue": "arXiv",
  "versions": []
}