{
  "abstract": "We study the long-time dynamics of small solutions to the one-dimensional nonlinear Schrödinger equation \\[ i\\partial_t v+\\partial_x^2v-β\\abs{v}^2v +\\cW(x)\\abs{v}^4v+γi\\partial_xv=0, \\] where $\\cW$ is spatially localized. The cubic nonlinearity is long range and produces the logarithmic phase correction, whereas the localized quintic term is short range at leading order. We construct a global forward modified wave operator for small complex asymptotic profiles and prove quantitative final-state estimates. For small data in the weighted energy space, we also establish global existence, sharp $t^{-1/2}$ decay, and forward modified scattering with a unique asymptotic profile. The principal new phenomenon occurs beyond this leading law. The exact Duhamel tail generated by the localized quintic term admits a quantitative inner scaling limit at the distinguished frequency $ζ=-γ/2$ on the scale $\\abs{ζ+γ/2}\\sim t^{-1/2}$. Its universal shape is explicit and depends on the value of the scattering profile on the distinguished ray and on the zeroth moment of $\\cW$. When both quantities are nonzero, the limit is nontrivial, belongs optimally to $C^{2,1}_{\\mathrm{loc}}$, and is not $C^3$ at the center. Away from the corresponding self-similar ray $ξ=-γ$, we construct rigorously defined higher-order outer expansions through every integer order allowed by the decay of $\\cW$, and to each fixed finite order when $\\cW$ is rapidly decreasing.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07044",
  "author_total": 3,
  "authors": [
    {
      "name": "Vinh Nguyen"
    },
    {
      "name": "Avy Soffer"
    },
    {
      "name": "Truong Vu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zej6qjdv66ty.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07044",
  "fulltext_url": null,
  "id": "7zej6qjdv66ty",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07044",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7212002614",
  "pdf_url": "https://arxiv.org/pdf/2609.07044",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zej6qjdv66ty.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07044",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07044"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Modified Scattering and Asymptotics for Perturbed One-Dimensional Cubic NLS",
  "title_text": "Modified Scattering and Asymptotics for Perturbed One-Dimensional Cubic NLS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/7zej6qjdv66ty",
  "venue": "arXiv",
  "versions": []
}