{
  "abstract": "We study the initial-boundary value problem for the Short Wave equation on the half-line $x\\ge 0$. A distinctive feature of this problem is that the boundary $x=0$ cannot be characterized \\emph{a priori} as an inflow or outflow boundary: its character is determined dynamically by the sign of the unknown trace $u(0,t)$. This leads to different analyticity properties of the associated eigenfunctions and, consequently, to different spectral formulations in the regimes $u(0,t)\\le0$ and $u(0,t)\\ge0$. Using the Unified Transform Method (aka the Fokas method), we formulate the solution in terms of matrix Riemann--Hilbert problems. We construct the associated spectral functions, derive the global relations, and show how the solution is reconstructed from the corresponding Riemann--Hilbert problem. In the case $u(0,t)\\le0$, the solution is determined by the initial data alone (assuming an appropriate decay as $x\\to \\infty$), whereas for $u(0,t)\\ge0$, compatible boundary data are also required for the construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07881",
  "author_total": 3,
  "authors": [
    {
      "name": "Iryna Karpenko"
    },
    {
      "name": "Dmitry Shepelsky"
    },
    {
      "name": "Ilona Tylevna"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3q5szftywq9ej.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07881",
  "fulltext_url": null,
  "id": "3q5szftywq9ej",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07881",
  "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"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7212019450",
  "pdf_url": "https://arxiv.org/pdf/2609.07881",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3q5szftywq9ej.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07881",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07881"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Short Wave equation on the half-line by the Unified Transform Method",
  "title_text": "The Short Wave equation on the half-line by the Unified Transform Method",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/3q5szftywq9ej",
  "venue": "arXiv",
  "versions": []
}