{
  "abstract": "We study the spectral stability of soliton solutions of the parametrically driven and damped nonlinear Schrödinger (PDDNLS) equation with power-law nonlinearity $|Ψ|^{2κ}Ψ$ in $N$ dimensions. We identify two families of stationary solutions, $Ψ_\\pm$, whose existence is determined by two conditions: one involving the nonlinearity exponent~$κ$ and the spatial dimension~$N$, and another involving the forcing amplitude~$r$ and the damping coefficient~$ρ$. By linearising around these solutions and introducing a convenient parametrisation through~$\\varepsilon$, we reduce the stability problem to a dissipationless eigenvalue problem for two Schrödinger-type operators, $L_0$ and $L_1$. We prove that $Ψ_-$ is always unstable, while for $Ψ_+$, we carry out a complete spectral analysis by decomposing the problem into angular momentum sectors $\\mathcal{S}_l$. We show that modes with $l \\geq 2$ generate no instabilities and that no complex quadruplets arise on~$\\mathcal{S}_1$. We also derive an analytical instability criterion on $\\mathcal{S}_l$ by combining a Vakhitov--Kolokolov-type argument with a local analysis that determines the direction of the eigenvalue splitting near the origin. In addition, we uncover new phenomena at both small and large values of $κ$ that were not investigated in earlier one-dimensional studies. Numerical stability diagrams for $N = 1, 2, 3$ are presented.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00346",
  "author_total": 1,
  "authors": [
    {
      "name": "Fernando Carreño-Navas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/04h33yj1zgb0d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00346",
  "fulltext_url": null,
  "id": "04h33yj1zgb0d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00346",
  "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.MP"
    ],
    [
      "arxiv",
      "nlin.PS"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00346",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/04h33yj1zgb0d.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.00346",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00346"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.PS",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Stability of Solitons in the Parametrically Driven and Damped Nonlinear Schrödinger Equation in Arbitrary Dimension",
  "title_text": "Stability of Solitons in the Parametrically Driven and Damped Nonlinear Schrödinger Equation in Arbitrary Dimension",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/04h33yj1zgb0d",
  "venue": "arXiv",
  "versions": []
}