{
  "abstract": "We construct modified wave operators for the nonlinear Schrödinger equation $i\\partial_tu+\\frac12Δu=|u|^pu$ in the full subcritical long-range case $0<p<2/d$, with small, nonvanishing, analytic final data $W(x)$ with bounded logarithmic gradients. Previous results established large-time asymptotics for selected classes of Cauchy data. Moreover, the exact asymptotic expansion for $p<1/d$ remained unknown. When $1/d<p<2/d$, our result gives the approximation $$\\frac{1}{(it)^{\\frac{d}{2}}} e^{\\frac{i|x|^2}{2t}} W\\left(\\frac{x}{t}\\right) \\exp\\left[ -i\\frac{t^{1-\\frac{dp}{2}}-1}{1-\\frac{dp}{2}} \\left|W\\left(\\frac{x}{t}\\right)\\right|^p \\right]$$ The wave operator is constructed by an iteration in the analytic spaces with decreasing radius. When $p\\le 1/d$, we construct the profile from a finite truncation of a Fuchsian equation coupled with a transport equation. This profile still leaves a long-range triangular coupling whose terminal integral does not preserve the required fast decay class. The construction yields quantitative $L^q$ asymptotics for $2\\le q\\le\\infty$ and uniqueness in the prescribed analytic asymptotic classes. The central new ingredients are a nonlinear final-state normal form that removes this long-range coupling and a mixed iteration in particular analytic spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11095",
  "author_total": 2,
  "authors": [
    {
      "name": "Jia Shen"
    },
    {
      "name": "Yifei Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0pfmx0f3998kq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11095",
  "fulltext_url": null,
  "id": "0pfmx0f3998kq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11095",
  "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.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7212225777",
  "pdf_url": "https://arxiv.org/pdf/2609.11095",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0pfmx0f3998kq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:26Z",
      "native_id": "2609.11095",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11095"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Modified wave operators for nonlinear Schrödinger equations in the full subcritical long range regime",
  "title_text": "Modified wave operators for nonlinear Schrödinger equations in the full subcritical long range regime",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/0pfmx0f3998kq",
  "venue": "arXiv",
  "versions": []
}