{
  "abstract": "Hirota's three-soliton condition is the standard test for the integrability of a bilinear equation $P(D)f\\cdot f=0$, and the folklore implication ``three-soliton condition $\\Rightarrow$ $N$-soliton solutions for all $N$'' underlies the classifications of Hietarinta. We study this implication in two directions. The first is the family $L\\,(D_x^3+Λ)^{2n+1}f\\cdot f=0$ with $L$, $Λ$ linear forms in the Hirota operators, which for $n=0$ contains the Korteweg--de Vries, Ito and Boiti--Leon--Manna--Pempinelli equations and for $L=D_x$ is the family of Gürses and Pekcan. On the principal component of the dispersion variety the $N$-soliton condition reduces to a sign sum of two bialternants; counting odd exponents shows that for $n=0$ the class is Hirota integrable and that for $n\\ge1$ the three-soliton condition holds for every $L$, and an explicit formula for the four-soliton residual proves the Gürses--Pekcan conjecture that the four-soliton condition fails for every $n\\ge1$. The formula yields sum-of-squares certificates excluding four-soliton solutions at every real configuration for the Gürses--Pekcan and Ito equations, classifies the real members of the class, and determines maximal numbers of nondegenerate solitons: exactly four or five for $n=1$ according as $L$ lies in the span of $D_x,Λ$ or not, five for $n=2$ and $L=D_x$, and finite for every $n\\ge1$ and every $L$. The second direction is a reformulation of the implication for a single non-hyperplane component of multiplicity one, which we prove, with complete classifications, for all graph-type dispersions $L(q(D)+D_t)$, for every irreducible even quadric component in any dimension and with an arbitrary cofactor, for parabolic components, for the dispersions $F(\\ell)+Q$ containing the Kadomtsev--Petviashvili surface, and, with a prescribed one-parameter phase family, for the cubic graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04953",
  "author_total": 1,
  "authors": [
    {
      "name": "Shi Zhenhua"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x1a53q49050j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04953",
  "fulltext_url": null,
  "id": "0x1a53q49050j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04953",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04953",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x1a53q49050j.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:25Z",
      "native_id": "2611.04953",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04953"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Multiplicity and parity in Hirota's $N$-soliton conditions: the bilinear families $L\\,(D_x^{3}+Λ)^{2n+1}$",
  "title_text": "Multiplicity and parity in Hirota's N-soliton conditions: the bilinear families L (D_x^3+Λ)^2n+1",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/0x1a53q49050j",
  "venue": "arXiv",
  "versions": []
}