{
  "abstract": "We study the one-dimensional periodic nonlinear Schrödinger equation with a power-type nonlinear damping, both in the focusing and the defocusing regime. For a sufficiently small damping parameter and initial data close in relative $H^1$ norm to a nonzero spatially homogeneous state, we prove global existence and asymptotic convergence, after a constant phase adjustment, to an explicit homogeneous solution. The \\(L^2\\) and \\(H^1\\) norms follow the explicit adiabatic decay law, with the same leading coefficient as the homogeneous solution. We obtain explicit convergence rates, including a \\(t^{-1/2}\\) bound for the relative spatial perturbation. For every nonconstant initial datum in a small neighborhood of the homogeneous solution, the kinetic energy has a strictly positive leading coefficient and decays one power of time faster than the squared \\(L^2\\) norm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16526",
  "author_total": 1,
  "authors": [
    {
      "name": "Boris Shakarov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7226mbdb8w28n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16526",
  "fulltext_url": null,
  "id": "7226mbdb8w28n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16526",
  "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": "W7221807647",
  "pdf_url": "https://arxiv.org/pdf/2610.16526",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7226mbdb8w28n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16526",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16526"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Asymptotic stability and sharp decay near homogeneous solutions of a nonlinearly damped Schrödinger equation on the torus",
  "title_text": "Asymptotic stability and sharp decay near homogeneous solutions of a nonlinearly damped Schrödinger equation on the torus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/7226mbdb8w28n",
  "venue": "arXiv",
  "versions": []
}