{
  "abstract": "We study a class of periodic NLS-Dysthe (NLS-Dy) equations, also referred to as higher-order nonlinear Schrödinger equations (HNLS), of the form $i\\partial_t u+a\\partial_x^2u+b|u|^2u+i\\varepsilon(c\\partial_x^3u+d|u|^2\\partial_xu+eu^2\\partial_x\\overline{u})=0$ with $ x\\in\\mathbb T$, $a,b,c,d,e\\in\\mathbb R$, $abc\\neq0$, and $\\varepsilon>0$. For $0<\\varepsilon\\ll1$, equations of this type arise as higher-order modulation models for some nonlinear dispersive PDEs, including the Klein-Gordon equation and the Serre-Green-Naghdi equation modeling the evolution of nonlinear shallow water waves. We prove local well-posedness in $H^s(\\mathbb T)$ for $s>3/2$ for arbitrary admissible coefficients, and establish an $H^1$ continuation criterion for $s\\ge2$. In the parameter regime $e\\neq0$, a lower-order conservation law provides uniform $H^1$ control and, together with the continuation criterion, yields global well-posedness for $s\\ge2$. This result does not require the non-resonance condition appearing in Takaoka's periodic theory, while the condition $e\\neq0$ naturally arises in the modulation setting considered here. Finally, motivated by the long-time regime relevant for modulation, we establish uniform long-time control without imposing a smallness assumption on the initial datum. Using the conservation laws of cubic NLS and their invariance under the commuting complex mKdV flow, we obtain uniform $H^s$ bounds on time intervals of order $\\varepsilon^{-1}$ for integer $s\\ge2$. For non-integer real $s\\ge2$, the maximal $H^s$ solution exists on the same time scale, while uniform control is obtained at a lower regularity, with a loss of less than one derivative.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14786",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrea Belloni"
    },
    {
      "name": "Filippo Giuliani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zq0zmb4vf8t4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14786",
  "fulltext_url": null,
  "id": "7zq0zmb4vf8t4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14786",
  "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": "W7221833764",
  "pdf_url": "https://arxiv.org/pdf/2610.14786",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zq0zmb4vf8t4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.14786",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14786"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Global Well-Posedness and Uniform Long-Time Control for a Class of Periodic NLS-Dysthe Equations",
  "title_text": "Global Well-Posedness and Uniform Long-Time Control for a Class of Periodic NLS-Dysthe Equations",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/7zq0zmb4vf8t4",
  "venue": "arXiv",
  "versions": []
}