{
  "abstract": "We study travelling front solutions of a family of degenerate Fisher-KPP equations $u_t=(u^n u_x)_x+u(1-u^n)$, where $n$ is a positive integer. At the minimal wave speed $c_{\\rm min}=\\tfrac{1}{\\sqrt{1+n}}$, these equations admit sharp front solutions, corresponding to an explicit heteroclinic orbit in the travelling wave phase-space. We show how this sharp front is perturbed when the wave speed is increased to $c=\\tfrac{1}{\\sqrt{1+n}}+\\varepsilon$, with $\\varepsilon$ sufficiently small. We use a well-motivated geometric desingularisation (also known as blow-up) near the degenerate equilibrium at the leading edge. By analysing the resulting directional and rescaling charts, we construct a singular heteroclinic orbit connecting the relevant asymptotic states, providing a simple geometric proof of the transition from sharp to smooth travelling fronts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23489",
  "author_total": 1,
  "authors": [
    {
      "name": "Zak Sattar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f5jfp014c4ga3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23489",
  "fulltext_url": null,
  "id": "f5jfp014c4ga3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23489",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7204208064",
  "pdf_url": "https://arxiv.org/pdf/2608.23489",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f5jfp014c4ga3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.23489",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23489"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Geometric desingularisation of the sharp-to-smooth travelling wave transition",
  "title_text": "Geometric desingularisation of the sharp-to-smooth travelling wave transition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/f5jfp014c4ga3",
  "venue": "arXiv",
  "versions": []
}