{
  "id": "2308.06655",
  "version": "v1",
  "published": "2023-08-13T00:21:04.000Z",
  "updated": "2023-08-13T00:21:04.000Z",
  "title": "Spectral and linear stability of peakons in the Novikov equation",
  "authors": [
    "Stéphane Lafortune"
  ],
  "comment": "19 pages, no figure",
  "categories": [
    "math.AP",
    "nlin.SI"
  ],
  "abstract": "The Novikov equation is a peakon equation with cubic nonlinearity which, like the Camassa-Holm and the Degasperis-Procesi, is completely integrable. In this article, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in $L^2(\\mathbb{R})$. To do so, we start with a linearized operator defined on $H^1(\\mathbb{R})$ and extend it to a linearized operator defined on weaker functions in $L^2(\\mathbb{R})$. The spectrum of the linearized operator in $L^2(\\mathbb{R})$ is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on $W^{1,\\infty}(\\mathbb{R})$ and linearly and spectrally stable on $H^1(\\mathbb{R})$. The result on $W^{1,\\infty}(\\mathbb{R})$ are in agreement with previous work about linear stability, while our results on $H^1(\\mathbb{R})$ are in agreement with the orbital stability obtained previously.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-13T00:21:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35C08",
      "35Q35"
    ],
    "keywords": [
      "linear stability",
      "novikov equation",
      "linearized operator",
      "cubic nonlinearity",
      "complex plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}