{
  "annotation_id": "f94f0d2d-40b6-489c-a8ea-2ee8b29cd319",
  "date_created": "2026-03-02T18:00:29.394000Z",
  "date_modified": "2026-03-02T18:00:29.394000Z",
  "file_hash": "26b0b81d1e2333f28c9cfef8ecffc8a75556df1ef8228319d6e8eab06540edf7",
  "private": false,
  "record": {
    "abstract": "We consider the simplest instabilities involving multiple unstable\nelectrostatic plasma waves corresponding to four-dimensional systems of mode\namplitude equations. In each case the coupled amplitude equations are derived\nup to third order terms. The nonlinear coefficients are singular in the limit\nin which the linear growth rates vanish together. These singularities are\nanalyzed using techniques developed in previous studies of a single unstable\nwave. In addition to the singularities familiar from the one mode problem,\nthere are new singularities in coefficients coupling the modes. The new\nsingularities are most severe when the two waves have the same linear phase\nvelocity and satisfy the spatial resonance condition $k_2=2k_1$. As a result\nthe short wave mode saturates at a dramatically smaller amplitude than that\npredicted for the weak growth rate regime on the basis of single mode theory.\nIn contrast the long wave mode retains the single mode scaling. If these\nresonance conditions are not satisfied both modes retain their single mode\nscaling and saturate at comparable amplitudes.",
    "arxiv_id": "patt-sol/9708004",
    "authors": [
      "John David Crawford",
      "Edgar Knobloch"
    ],
    "categories": [
      "patt-sol",
      "nlin.PS"
    ],
    "doi": "10.1017/S0022377898006540",
    "title": "Amplitude equations for coupled electrostatic waves in the limit of weak   instability",
    "url": "https://arxiv.org/abs/patt-sol/9708004"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "694eafdb-7f7d-485a-a22f-eb975cf88ed2",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}