{
  "abstract": "We develop a matrix pseudodifference operator framework that places theAdler-Pfaff and Pfaff-Toda hierarchies in a common algebraic setting. The bi-infinite Adler-Pfaff hierarchy then becomes a $2\\times2$ matrix pseudodifference Lax hierarchy. We introduce the Matrix Pfaff-Toda hierarchy by dressing the matrix Laurent algebra $M_{2N}\\bigl(\\mathbb{C}[\\mathcal{S},\\mathcal{S}^{-1}]\\bigr)$ using a Pfaff-Toda splitting of the matrix pseudodifference algebra. In the one-component case, its diagonal sector recovers the continuous Pfaff-Toda hierarchy, while the off-diagonal directions supply the missing odd flows, and the powers of a single bare operator reconstruct the full Adler-Pfaff hierarchy. For general $N$, on the regular factorization locus, the multicomponent Pfaff-Toda tau-functions of Savchenko and Zabrodin realize the commuting diagonal sector, with its dressing equations derived directly from the fermionic bilinear identity. Reductions to the block-diagonal and scalar cases recover multicomponent and scalar $2D$ Toda, identifying the even Pfaff hierarchy with an anti-diagonal Toda subhierarchy, and yielding the Krichever-Zabrodin C-Toda hierarchy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12970",
  "author_total": 2,
  "authors": [
    {
      "name": "Sylvain Carpentier"
    },
    {
      "name": "Marta Dell'Atti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fhfpq0xdvwc28.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12970",
  "fulltext_url": null,
  "id": "fhfpq0xdvwc28",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12970",
  "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",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7212995759",
  "pdf_url": "https://arxiv.org/pdf/2609.12970",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fhfpq0xdvwc28.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:07Z",
      "native_id": "2609.12970",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12970"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Matrix Dressing Beyond Pfaff-Toda",
  "title_text": "Matrix Dressing Beyond Pfaff-Toda",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/fhfpq0xdvwc28",
  "venue": "arXiv",
  "versions": []
}