{
  "abstract": "We prove long-time strong Sobolev instability for a class of quasilinear Schrödinger equations on $\\mathbb T^2$ arising, via the Madelung transform, in the description of Euler-Korteweg capillary fluids. More precisely, for every $s\\ge 7$, arbitrarily small $μ>0$, and arbitrarily large $\\mathcal K>0$, we construct a solution such that $$ \\|u(0)\\|_{H^s}<μ, \\qquad \\|u(T)\\|_{H^s}>\\mathcal K $$ at some finite time $T$, for which we provide an explicit exponential upper bound. The instability mechanism is generated by a finite-dimensional resonant Toy Model associated with the cubic semilinear part of the equation, in the spirit of Colliander-Keel-Staffilani-Takaoka-Tao. The main technical difficulty is to justify this dynamics over the long time scale of the energy transfer in the presence of quasilinear derivative losses. We achieve this by combining a weak Birkhoff normal form with paradifferential calculus, microlocal symmetrization, and modified energy estimates. Our result provides a rigorous quasilinear realization of a resonant energy-transfer mechanism, and suggests a connection with weakly turbulent dynamics in capillary fluids.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37593",
  "author_total": 4,
  "authors": [
    {
      "name": "Yuri Cacchiò"
    },
    {
      "name": "Filippo Giuliani"
    },
    {
      "name": "Felice Iandoli"
    },
    {
      "name": "Raffaele Scandone"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eg2s826vr3rx3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37593",
  "fulltext_url": null,
  "id": "eg2s826vr3rx3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37593",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7214958747",
  "pdf_url": "https://arxiv.org/pdf/2609.37593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eg2s826vr3rx3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Long time strong Sobolev instability for quasilinear NLS",
  "title_text": "Long time strong Sobolev instability for quasilinear NLS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/eg2s826vr3rx3",
  "venue": "arXiv",
  "versions": []
}