{
  "abstract": "We study an inverse source problem for a two-mode nonlinear Helmholtz system motivated by second-harmonic generation. The datum is the full first Fréchet derivative of the nonlinear Dirichlet-to-Neumann map at one fixed, common, small boundary state, which need not be zero. For real-valued small data, a known susceptibility bounded away from zero, and sources supported in a fixed compact subset of the domain, we prove uniqueness and a conditional single-logarithmic stability estimate. The first boundary variation is the Dirichlet-to-Neumann map of a symmetric $2\\times2$ matrix Schrödinger operator. Absorbing the two known Helmholtz energies into its matrix potential permits the use of standard zero-energy complex geometrical optics solutions with a common null phase geometry. A bilinear Alessandrini identity then gives a uniform Fourier estimate for the two entries containing the background fields. Zero extension below the exponent $3/2$, a low/high frequency splitting, and interpolation between $H^{-2}$ and the a priori $H^s$ source bound yield the stated stability modulus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09845",
  "author_total": 3,
  "authors": [
    {
      "name": "Hongcan Ye"
    },
    {
      "name": "Lei Zhang"
    },
    {
      "name": "Ting Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9kyttrj5tye4p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09845",
  "fulltext_url": null,
  "id": "9kyttrj5tye4p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09845",
  "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|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7212124661",
  "pdf_url": "https://arxiv.org/pdf/2609.09845",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9kyttrj5tye4p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09845",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09845"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Logarithmic Stability for an Inverse Source Problem in a Coupled Nonlinear Helmholtz System",
  "title_text": "Logarithmic Stability for an Inverse Source Problem in a Coupled Nonlinear Helmholtz System",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/9kyttrj5tye4p",
  "venue": "arXiv",
  "versions": []
}