{
  "id": "2006.03345",
  "version": "v1",
  "published": "2020-06-05T10:02:12.000Z",
  "updated": "2020-06-05T10:02:12.000Z",
  "title": "Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity",
  "authors": [
    "Nabile Boussaid",
    "Claudio Cacciapuoti",
    "Raffaele Carlone",
    "Andrew Comech",
    "Diego Noja",
    "Andrea Posilicano"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the $\\mathbf{SU}(1,1)$-symmetry: the first preserving and the second breaking the parity symmetry. We show that a perturbation which breaks the $\\mathbf{SU}(1,1)$-symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a perturbation which breaks both the $\\mathbf{SU}(1,1)$-symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. The developing instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues $\\pm 2\\omega\\mathrm{i}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-05T10:02:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral stability",
      "concentrated nonlinearity",
      "parity symmetry",
      "instability",
      "weakly relativistic solitary waves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}