{
  "annotation_id": "a99a54ff-372f-4921-b0d3-185c4c341abd",
  "date_created": "2026-03-02T18:00:29.398000Z",
  "date_modified": "2026-03-02T18:00:29.398000Z",
  "file_hash": "b6411bea3f271e2c2121875031741deddec40365f0082e72bddd08babe6d8934",
  "private": false,
  "record": {
    "abstract": "We determine the modulational stability of standing waves with small group\nvelocity in quasi-onedimensional systems slightly above the threshold of a\nsupercritical Hopf bifurcation. The stability limits are given by two different\nlong-wavelength destabilisation mechanisms and generically also a\nshort-wavelength destabilisation. The Eckhaus parabola is shifted off-center\nand can be convex from below or above. For nonzero group velocity the Newell\ncriterion, which near the cross-over from standing to traveling waves becomes a\nrather weak condition, does not determine the destabilisation of all standing\nwaves in one dimension. The cross-over to the non-local equations that are\nasymptotically valid near threshold is discussed in detail. Close to the\ntransition from standing to traveling waves complex dynamics can arise due to\nthe competition of counter-propagating waves and the wavenumber selection by\nsources. Our results yield necessary conditions for the stability of traveling\nrectangles in quasi-twodimensional systems with axial anisotropy and form a\nstarting point for understanding the spatio-temporal chaos of traveling oblique\nrolls observed in electroconvection of nematic liquid crystals.",
    "arxiv_id": "patt-sol/9805005",
    "authors": [
      "Hermann Riecke",
      "Lorenz Kramer"
    ],
    "categories": [
      "patt-sol",
      "nlin.PS"
    ],
    "title": "The Stability of Standing Waves",
    "url": "https://arxiv.org/abs/patt-sol/9805005"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "5167bef8-1ade-44c1-9b14-04694ae2daf1",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}