{
  "abstract": "Fu and Nijhoff's direct linearization scheme for the KP hierarchy and its reductions employs a system of evolution equations of an infinite matrix $U$ with quadratic nonlinearity. Part of the matrix elements of $U$ can be identified with affine coordinates $w_{ij}$ of the top cell of the Sato Grassmannian. The evolution equations of these matrix elements are identical to the evolution equations of $w_{ij}$ representing the KP hierarchy in geometric terms. This geometric interpretation can be extended to other matrix elements of $U$ by introducing negative flows to the evolution equations of $U$. The extended system turns out to be substantially equivalent to the two-component KP hierarchy. The Cauchy matrix approach to the KP hierarchy can be explained in this geometric perspective. Multi-component generalizations of Fu and Nijhoff's nonlinear system are related to the AKNS and ASDYM hierarchies. Multi-component Sato Grassmannians show up therein as the relevant geometric structure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24538",
  "author_total": 1,
  "authors": [
    {
      "name": "Kanehisa Takasaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3m6d57kcdf68e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24538",
  "fulltext_url": null,
  "id": "3m6d57kcdf68e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24538",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7204275242",
  "pdf_url": "https://arxiv.org/pdf/2608.24538",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3m6d57kcdf68e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2608.24538",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24538"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Direct linearization, Cauchy matrix and Sato Grassmannian",
  "title_text": "Direct linearization, Cauchy matrix and Sato Grassmannian",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/3m6d57kcdf68e",
  "venue": "arXiv",
  "versions": []
}