{
  "annotation_id": "15bd5518-b6b6-4982-b645-698251ff135a",
  "date_created": "2026-03-02T18:00:29.152000Z",
  "date_modified": "2026-03-02T18:00:29.152000Z",
  "file_hash": "9ebd6609fd6ed11d618bb3d92f511743f047222e23bcf7544a51b7925df0a5bf",
  "private": false,
  "record": {
    "abstract": "In this paper we describe invariant geometrical ~structures in the phase\nspace of the Swift-Hohenberg equation in a neighborhood of its periodic\nstationary states. We show that in spite of the fact that these states are only\nmarginally stable (i.e., the linearized problem about these states has\ncontinuous spectrum extending all the way up to zero), there exist finite\ndimensional invariant manifolds in the phase space of this equation which\ndetermine the long-time behavior of solutions near these stationary solutions.\nIn particular, using this point of view, we obtain a new demonstration of\nSchneider\u0027s recent proof that these states are nonlinearly stable.",
    "arxiv_id": "patt-sol/9612002",
    "authors": [
      "J. -P. Eckmann",
      "C. E. Wayne",
      "P. Wittwer"
    ],
    "categories": [
      "patt-sol",
      "chao-dyn",
      "funct-an",
      "math.FA",
      "nlin.CD",
      "nlin.PS",
      "physics.flu-dyn"
    ],
    "doi": "10.1007/s002200050238",
    "title": "Geometric Stability Analysis for Periodic Solutions of the   Swift-Hohenberg Equation",
    "url": "https://arxiv.org/abs/patt-sol/9612002"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "992d3cfd-bb46-470c-831a-1c300d80016a",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}