{
  "abstract": "We investigate the long-time behavior of small solutions to the one-dimensional Hartree equation with the soft Coulomb interaction. The Hartree equation arises as an effective mean-field model for the evolution of many-body quantum systems, while the soft Coulomb potential provides a regularized one-dimensional approximation of the Coulomb interaction that removes the singularity at the origin while preserving its long-range character. Owing to the slow spatial decay of the interaction kernel, the equation exhibits long-range nonlinear effects. We prove global existence and modified scattering for sufficiently small, exponentially localized initial data. A notable feature is that, in contrast to the logarithmic phase correction arising in the well-known three-dimensional Coulomb Hartree equation, the one-dimensional soft Coulomb interaction produces a leading phase correction of order $(\\log t)^2$, followed by a lower-order logarithmic correction. The proof is carried out in analytic function spaces. The exponential localization of the initial data is converted into analyticity of the transformed profile, and a method based on a generator function with a time-dependent radius of analyticity is used to compensate for the derivative loss arising from the nonlinear phase equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31515",
  "author_total": 2,
  "authors": [
    {
      "name": "Changhun Yang"
    },
    {
      "name": "Chanjin You"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7wa6bnpw694xm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31515",
  "fulltext_url": null,
  "id": "7wa6bnpw694xm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31515",
  "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": "W7214617476",
  "pdf_url": "https://arxiv.org/pdf/2609.31515",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7wa6bnpw694xm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31515",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31515"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Asymptotic Behavior of One-Dimensional Hartree Equations with the Long-Range Interaction",
  "title_text": "Asymptotic Behavior of One-Dimensional Hartree Equations with the Long-Range Interaction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/7wa6bnpw694xm",
  "venue": "arXiv",
  "versions": []
}