{
  "abstract": "A hypercomplex method for constructing solutions of a class of nonlinear partial differential equations is developed. A family of two-dimensional commutative associative algebras is introduced in which the structural coefficients depend on a component of the monogenic function itself. A corresponding theory of monogenic functions is developed, and nonlinear analogues of the Cauchy-Riemann conditions are obtained. It is shown that the first component of a monogenic function satisfies a broad class of second-order nonlinear partial differential equations whose coefficients depend on the unknown function. A hodograph transformation is used to linearize the nonlinear Cauchy-Riemann system. As a result, the problem of constructing solutions of the nonlinear equations is reduced to solving a second-order linear partial differential equation with variable coefficients, followed by the inverse hodograph transformation. A local correspondence between solutions of the nonlinear Cauchy-Riemann system and solutions of the associated linear equation is established, yielding a constructive algorithm for generating solutions of the original nonlinear PDEs. Several examples are presented in which explicit and implicit solutions of nonlinear second-order PDEs are obtained by this method.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27042",
  "author_total": 1,
  "authors": [
    {
      "name": "Vitalii Shpakivskyi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0efktzs7jeqss.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27042",
  "fulltext_url": null,
  "id": "0efktzs7jeqss",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27042",
  "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",
      "math-ph"
    ],
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      "openalex",
      "2604|Algebraic and Geometric Analysis"
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  "openalex": "W7223115944",
  "pdf_url": "https://arxiv.org/pdf/2610.27042",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0efktzs7jeqss.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:04Z",
      "native_id": "2610.27042",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27042"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Hypercomplex Method for Solving Nonlinear PDEs",
  "title_text": "Hypercomplex Method for Solving Nonlinear PDEs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/0efktzs7jeqss",
  "venue": "arXiv",
  "versions": []
}