{
  "abstract": "In this article, we investigate the stochastic Camassa--Holm equation driven by additive noise, which models nonlinear shallow-water waves under random external forcing. We establish that, almost surely, any finite-time breakdown occurs through wave breaking: the wave amplitude and the positive part of the slope remain bounded, whereas the minimum slope tends to $-\\infty$. Under an additional moment assumption, we derive an initial-slope condition ensuring wave breaking with positive probability. We also formulate the equation as a stochastic Hamiltonian PDE, establish its stochastic multi-symplectic conservation law, and derive the evolution law for the averaged $H^1$ energy. Combining the Fourier--Galerkin method, symplectic Runge--Kutta integration, and exact solution of the linear stochastic subsystem, we propose a fully discrete method which preserves an integrated discrete stochastic multi-symplectic conservation law and the linear growth of the discrete averaged \\(H^1\\) energy. Under suitable regularity and resolution assumptions, we present that the fully discrete dynamics retain the slope-steepening mechanism responsible for wave breaking, providing a discrete counterpart of the continuous behavior.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32128",
  "author_total": 3,
  "authors": [
    {
      "name": "Liying Sun"
    },
    {
      "name": "Zizhao Sun"
    },
    {
      "name": "Liying Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e53g25qc1eay3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32128",
  "fulltext_url": null,
  "id": "e53g25qc1eay3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32128",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7214716647",
  "pdf_url": "https://arxiv.org/pdf/2609.32128",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e53g25qc1eay3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32128",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32128"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Wave Breaking and Structure-Preserving Fully Discrete Method for Stochastic Camassa--Holm Equation with Additive Noise",
  "title_text": "Wave Breaking and Structure-Preserving Fully Discrete Method for Stochastic Camassa--Holm Equation with Additive Noise",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/e53g25qc1eay3",
  "venue": "arXiv",
  "versions": []
}