{
  "abstract": "We establish stable cusp formation for the Novikov equation, a cubically nonlinear Camassa--Holm-type equation. We identify an open set of smooth initial data for which the first gradient blow-up produces a cusp with sharp Hölder regularity $C^{7/9}$. This result shows that, in nonlocal wave-breaking problems, the sharp regularity of the cusp is not determined by the nonlocal or nonlinear structure alone. While the conserved $H^1$-type quantity excludes the $C^{1/3}$ cusp associated with Burgers-type gradient blow-up, the precise Hölder exponent is selected by the coupling between the nonlocal term and the algebraic structure of the nonlinearity. In the Novikov equation, this coupling yields the exponent $7/9$, rather than the $3/5$ exponent known for the Camassa--Holm and Hunter--Saxton equations. The main difficulty is that the naive high-frequency limit retains the cubic character of the equation and therefore does not exhibit a self-similar leading flow. We overcome this by introducing a Galilean-type change of variables around a nonzero background, which reveals a quadratic Hunter--Saxton-type leading equation. Its self-similar profiles determine the $C^{7/9}$ cusp, while the nonlocal and cubic remainders are controlled perturbatively in modulated similarity variables.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29236",
  "author_total": 4,
  "authors": [
    {
      "name": "Yunjoo Kim"
    },
    {
      "name": "Dowan Koo"
    },
    {
      "name": "Bongsuk Kwon"
    },
    {
      "name": "Wanyong Shim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6my14nemvr1p9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29236",
  "fulltext_url": null,
  "id": "6my14nemvr1p9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29236",
  "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.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7204908586",
  "pdf_url": "https://arxiv.org/pdf/2608.29236",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6my14nemvr1p9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29236",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29236"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Stable $C^{7/9}$ cusp formation for the Novikov equation",
  "title_text": "Stable C^7/9 cusp formation for the Novikov equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/6my14nemvr1p9",
  "venue": "arXiv",
  "versions": []
}