{
  "abstract": "The inverse scattering problem for the one-dimensional nonlinear Dirac equation \\begin{align*} \\begin{cases} i(u_t+u_x)+v=\\mathcal{N}_1(u,v),\\\\ i(v_t-v_x)+u=\\mathcal{N}_2(u,v) \\end{cases} \\end{align*} is studied. We assume that the unknown nonlinearities $\\mathcal{N}_j(u,v)$ ($j=1,2$) of the equation belong to $C^\\infty(\\mathbb{C}^2;\\mathbb{C})$ and satisfy $(\\partial_\\mathbf{z}^α\\mathcal{N}_j)(\\mathbf{z})=O(|\\mathbf{z}|^{\\max\\{ 5-|α|,0 \\}})$ ($|\\mathbf{z}|\\to 0$) for any multi-index $α\\in (\\mathbb{N} \\cup \\{ 0 \\})^4$. Here, $\\mathbf{z}=(z_1,z_2)\\in \\mathbb{C}^2$, $ \\partial_\\mathbf{z}^α= (\\partial_{z_1},\\partial_{\\overline{z_1}},\\partial_{z_2},\\partial_{\\overline{z_2}})^α, $ and $\\partial_{z_1},\\partial_{\\overline{z_1}},\\partial_{z_2},\\partial_{\\overline{z_2}}$ are the Wirtinger differential operators. Under some additional assumptions on $\\mathcal{N}_j$, we establish a reconstruction formula for $(\\partial_\\mathbf{z}^α\\mathcal{N}_j)(0)$ ($|α|\\ge 5$) using the knowledge of the scattering operator for the equation. Although such results were established for the nonlinear Schrödinger equation [Sasaki 2024] and the nonlinear Klein-Gordon equation [Sasaki 2025], the methods in [Sasaki 2024, 2025] are not directly applicable to the nonlinear Dirac equation due to its system nature. Specifically, it is quite difficult to prove the invertibility of a matrix associated with the input data. To overcome this difficulty, we introduce a new method based on the massless limit for the free solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23628",
  "author_total": 1,
  "authors": [
    {
      "name": "Hironobu Sasaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8345rqpd1ynq9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23628",
  "fulltext_url": null,
  "id": "8345rqpd1ynq9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23628",
  "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": "W7213997071",
  "pdf_url": "https://arxiv.org/pdf/2609.23628",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8345rqpd1ynq9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23628",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23628"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On inverse scattering for the one-dimensional nonlinear Dirac equation",
  "title_text": "On inverse scattering for the one-dimensional nonlinear Dirac equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8345rqpd1ynq9",
  "venue": "arXiv",
  "versions": []
}