{
  "annotation_id": "52b6434e-b9e7-458e-b2e3-554730ae272c",
  "date_created": "2026-03-02T18:00:28.874000Z",
  "date_modified": "2026-03-02T18:00:28.874000Z",
  "file_hash": "0a4521666ab2912440fd0c77987431dcf0008e2914a022c32a9c5bade7eb0fd0",
  "private": false,
  "record": {
    "abstract": "Motivated by experimental observations of exotic standing wave patterns in\nthe two-frequency Faraday experiment, we investigate the role of normal form\nsymmetries in the pattern selection problem. With forcing frequency components\nin ratio m/n, where m and n are co-prime integers, there is the possibility\nthat both harmonic and subharmonic waves may lose stability simultaneously,\neach with a different wavenumber. We focus on this situation and compare the\ncase where the harmonic waves have a longer wavelength than the subharmonic\nwaves with the case where the harmonic waves have a shorter wavelength. We show\nthat in the former case a normal form transformation can be used to remove all\nquadratic terms from the amplitude equations governing the relevant resonant\ntriad interactions. Thus the role of resonant triads in the pattern selection\nproblem is greatly diminished in this situation. We verify our general results\nwithin the example of one-dimensional surface wave solutions of the\nZhang-Vinals model of the two-frequency Faraday problem. In one-dimension, a\n1:2 spatial resonance takes the place of a resonant triad in our investigation.\nWe find that when the bifurcating modes are in this spatial resonance, it\ndramatically effects the bifurcation to subharmonic waves in the case of\nforcing frequencies are in ratio 1/2; this is consistent with the results of\nZhang and Vinals. In sharp contrast, we find that when the forcing frequencies\nare in ratio 2/3, the bifurcation to (sub)harmonic waves is insensitive to the\npresence of another spatially-resonant bifurcating mode.",
    "arxiv_id": "patt-sol/9810003",
    "authors": [
      "Mary Silber",
      "Anne C. Skeldon"
    ],
    "categories": [
      "patt-sol",
      "nlin.PS"
    ],
    "doi": "10.1103/PhysRevE.59.5446",
    "title": "Parametrically Excited Surface Waves: Two-Frequency Forcing, Normal Form   Symmetries, and Pattern Selection",
    "url": "https://arxiv.org/abs/patt-sol/9810003"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "d7809061-ebc9-464d-9e5b-dd604e511cda",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}