{
  "abstract": "We investigate the long-time asymptotics of a full soliton gas for the Camassa--Holm equation. The analysis starts from a pure-soliton Riemann--Hilbert (RH) problem with \\(2N\\) poles and two distinct types of residue conditions. We prove that, as \\(N\\to\\infty\\), this discrete RH problem converges to a limiting soliton gas RH problem whose jump matrix contains two nonzero reflection coefficients. In this sense, the limiting problem gives a full soliton gas model for the Camassa--Holm equation, in contrast to the previously studied half soliton gas models, whose jump matrices involve only one nonzero reflection coefficient. The limiting RH problem is analyzed by the Deift--Zhou nonlinear steepest descent method. The presence of two nonzero reflection coefficients requires two different types of triangular factorizations of the jump matrix and leads to a more delicate \\(g\\)-function mechanism. The main difficulty lies in the construction of suitable \\(g\\)-functions adapted to the Camassa--Holm phase, together with the precise control of their behavior near the distinguished point \\(k=i/2\\) and at infinity. Depending on the location of the spectral endpoints \\(η_1\\) and \\(η_2\\), different \\(g\\)-function mechanisms arise. In this paper, we focus on Case I and derive the long-time asymptotic formulas in three elliptic-wave regions of the self-similar plane. In each region, the leading term is given by a finite-gap elliptic function, while in the central region the first correction is of order \\(\\mathcal O(t^{-1/2})\\) and involves parabolic cylinder functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.10727",
  "author_total": 3,
  "authors": [
    {
      "name": "Dedi Yan"
    },
    {
      "name": "Xianguo Geng"
    },
    {
      "name": "Minxin Jia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3rkv1pxzm7n6y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.10727",
  "fulltext_url": null,
  "id": "3rkv1pxzm7n6y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.10727",
  "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"
    ],
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7164160566",
  "pdf_url": "https://arxiv.org/pdf/2606.10727",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3rkv1pxzm7n6y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2606.10727",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.10727"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Long-time asymptotics of a full Camassa-Holm soliton gas",
  "title_text": "Long-time asymptotics of a full Camassa-Holm soliton gas",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/3rkv1pxzm7n6y",
  "venue": "arXiv",
  "versions": []
}