{
  "abstract": "We study simultaneous recovery of the (variable) mass density, initial displacement, and initial velocity for the three-dimensional isotropic elastic wave equation with known constant Lamé parameters. The data are the complete displacement trace on an enclosing boundary. We first assume that the density-weighted initial displacement and velocity have fixed known profiles in one spatial direction. The $s^0$ and $s^1$ coefficients of the zero-frequency Laplace expansion identify these weighted states. The $s^2$ and $s^3$ coefficients then give static Lamé orthogonality identities for the density difference. The exact difference expansion through order $s^3$ has an $O(s^4)$ remainder, uniformly on bounded spatial sets and bounded density-contrast and weighted-state classes. Under alignment of the two initial states and a nonzero moment of the density-weighted initial velocity, the two density identities reduce to a constant-vector static transform. Two opposite elastic null phases give uniqueness for one or two fixed vertical density profiles when the profile family is two-sided Laplace nondegenerate. When $λ+μ\\ne0$, each nonzero normal root has partial multiplicities $2$ and $1$ and admits a length-two Jordan chain. The associated polynomial--exponential Lamé mode produces derivatives of the bilateral profile transforms. The resulting Hermite--Laplace system gives uniqueness for aligned classes with up to four fixed vertical profiles and independent horizontal coefficients, provided that the profile system is Hermite--Laplace nondegenerate. Distinct translations of one compactly supported profile provide an explicit four-profile class. We also prove rigidity of the alignment reduction and exhibit an infinite-dimensional kernel for the reduced transform on unrestricted densities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14215",
  "author_total": 3,
  "authors": [
    {
      "name": "Yixian Gao"
    },
    {
      "name": "Hongyu Liu"
    },
    {
      "name": "Yang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/af0yrk7a5558p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14215",
  "fulltext_url": null,
  "id": "af0yrk7a5558p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14215",
  "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|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7213241149",
  "pdf_url": "https://arxiv.org/pdf/2609.14215",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/af0yrk7a5558p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14215",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14215"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On passive recovery of structured elastic density and initial states",
  "title_text": "On passive recovery of structured elastic density and initial states",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/af0yrk7a5558p",
  "venue": "arXiv",
  "versions": []
}