{
  "abstract": "We compute the effective six-wave interaction in spatial and temporal Hamiltonian Dysthe equations and compare it with the complete degree-six coefficient of the deep-water Euler Hamiltonian. On every compact subset $K$ of a generic quadratic-dispersion resonance patch, exact resonances persist under a small cubic correction to the dispersion, and \\[ W_6\\bigl(Φ_{λ,\\varepsilon}(z)\\bigr) =\\varepsilon\\frac{4BD}{aL(z)}+O_K(\\varepsilon^2) \\] uniformly in the resonance point $z$ and the deformation parameter $λ$. Here $L$ is the spectral diameter and $B,D$ are the local cubic and mean-flow coefficients. Thus $BD\\ne0$ gives nonzero scattering on relatively open resonance sets. At the Craig--Guyenne--Sulem temporal coefficients, the corresponding exact Euler resonances satisfy \\[ \\fracμ{4}W_6^{\\rm E}\\bigl(1+μΨ_μ(z)\\bigr) =-\\frac{8}{L(z)}+O_K(μ), \\] where $μ$ is the relative bandwidth. The leading term comes from nine contractions of quartic vertices; every other connected degree-six contribution is uniformly bounded. We classify the coefficient values for which the generic six-wave coefficient vanishes identically; in the temporal case, this also gives cancellation on every non-permutation four-wave resonance. Outside these sets, the Zakharov--Schulman condition restricts every regular $C^3$ quadratic leading symbol to $\\operatorname{span}\\{1,k,ω(k)\\}$, the span of the symbols of wave action, momentum, and linear energy. Consequently the physical Dysthe models cannot support an inverse-scattering hierarchy with infinitely many linearly independent regular $C^3$ quadratic leading symbols. The cancellation sets contain known nonlinear Schrödinger, mixed Chen--Lee--Liu, Calogero--Moser derivative nonlinear Schrödinger, and Hirota representatives.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23175",
  "author_total": 2,
  "authors": [
    {
      "name": "Alex J. Sutherland"
    },
    {
      "name": "Solomon C. Yim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cg4ke58e626z2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23175",
  "fulltext_url": null,
  "id": "cg4ke58e626z2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23175",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7214029402",
  "pdf_url": "https://arxiv.org/pdf/2609.23175",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cg4ke58e626z2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.23175",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23175"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Six-wave scattering in Hamiltonian Dysthe equations: Euler matching and integrability obstructions",
  "title_text": "Six-wave scattering in Hamiltonian Dysthe equations: Euler matching and integrability obstructions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/cg4ke58e626z2",
  "venue": "arXiv",
  "versions": []
}