{
  "abstract": "We study the orbital instability of algebraically decaying stationary solutions of nonlinear Schrödinger equations in dimensions $d\\ge2$, taking competing double-power nonlinearities as a canonical model. At zero frequency, the essential spectrum of the linearized operator touches zero, and the usual $H^1$ coercivity is lost. We construct the zero-frequency limit $η_0$ of the frequency derivatives $\\partial_ωϕ_ω$ of standing-wave profiles and determine its sharp asymptotic behavior. For a stationary ground state $ϕ_0\\in H^1(\\mathbb{R}^d)$, we prove instability if $\\langle x\\rangle\\,ϕ_0\\notin L^2(\\mathbb{R}^d)$ or, when $\\langle x\\rangle\\,ϕ_0\\in L^2(\\mathbb{R}^d)$, if the duality pairing $\\langleη_0,ϕ_0\\rangle$ is negative. This criterion is a zero-frequency counterpart of the negative slope condition in the Grillakis--Shatah--Strauss theory. The proof combines variation of parameters for the linearized ODE with a shooting analysis that determines the global behavior of the homogeneous solution regular at the origin. We then localize $η_0$ to construct an unstable direction and establish instability through a Lyapunov functional argument based on the variational characterization of ground states. The results extend with essentially the same proofs to a class of multiple-power nonlinearities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39830",
  "author_total": 2,
  "authors": [
    {
      "name": "Noriyoshi Fukaya"
    },
    {
      "name": "Masayuki Hayashi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vy0xdcprtkr6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39830",
  "fulltext_url": null,
  "id": "5vy0xdcprtkr6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39830",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39830",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vy0xdcprtkr6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39830",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39830"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Instability theory of stationary solutions for nonlinear Schrödinger equations",
  "title_text": "Instability theory of stationary solutions for nonlinear Schrödinger equations",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5vy0xdcprtkr6",
  "venue": "arXiv",
  "versions": []
}