{
  "abstract": "It has been known that (the square of) Jacobi elliptic function can be expressed by an infinite sum of single solitons of $\\sech^2$ shape. In this paper, we show that there exists a similar structure for higher genus cases. It turns out that this is just a consequence of the quasi-periodicity of the Riemann $θ$-function. More precisely, we show that the second derivative of $\\log θ$ with the \\emph{real} Riemann $θ$-function of genus $g$ near soliton limit can be well approximated by the \\emph{sum} of \\emph{real} and \\emph{regular} $g$-soliton solutions of Hirota-type in $g$-dimensional real space $\\R^g$. This leads to a tessellation of $\\R^g$, whose tile is an oblique prism divided into $2^g$ sections by hyper-planes of dominant exponents in the theta function. We apply the results to study quasi-periodic solutions to the KdV and KP equations. We construct the quasi-periodic solutions using the Schottky group, which uniformizes the corresponding Riemann surfaces. We also discuss solitons on quasi-periodic background by pinching some of the homological cycles",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05716",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuji Kodama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0c1jqbzk7ftfa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05716",
  "fulltext_url": null,
  "id": "0c1jqbzk7ftfa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05716",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7220720690",
  "pdf_url": "https://arxiv.org/pdf/2610.05716",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0c1jqbzk7ftfa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2610.05716",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05716"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "The Riemann theta function near soliton limit",
  "title_text": "The Riemann theta function near soliton limit",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/0c1jqbzk7ftfa",
  "venue": "arXiv",
  "versions": []
}