{
  "id": "2402.08759",
  "version": "v2",
  "published": "2024-02-13T19:52:41.000Z",
  "updated": "2024-09-21T01:30:51.000Z",
  "title": "Spectral instability of peakons for the $b$-family of Novikov equations",
  "authors": [
    "Xijun Deng",
    "Stéphane Lafortune"
  ],
  "comment": "19 pages, no figure",
  "categories": [
    "math.AP",
    "nlin.SI"
  ],
  "abstract": "In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter $b$. This family is called the ``$b$-Novikov'' since it reduces to the integrable Novikov equation in the case $b=3$. By extending the corresponding linearized operator defined on functions in $H^1(\\mathbb{R})$ to one defined on weaker functions on $L^2(\\mathbb{R})$, we prove spectral and linear instability on $L^2(\\mathbb{R})$ of peakons in the $b$-Novikov equations for any $b$. We also consider the stability on $H^1(\\mathbb{R})$ and show that the peakons are spectrally or linearly stable only in the case $b=3$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-09-21T01:30:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35C08",
      "35P05",
      "35Q35"
    ],
    "keywords": [
      "spectral instability",
      "cubic nonlinearity",
      "integrable novikov equation",
      "peakon equations",
      "weaker functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}