{
  "id": "1203.3859",
  "version": "v2",
  "published": "2012-03-17T12:51:33.000Z",
  "updated": "2012-07-13T23:28:16.000Z",
  "title": "Linear instability of nonlinear Dirac equation in 1D with higher order nonlinearity",
  "authors": [
    "Andrew Comech"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.SP",
    "nlin.PS"
  ],
  "abstract": "We consider the nonlinear Dirac equation in one dimension, also known as the Soler model in (1+1) dimensions, or the massive Gross-Neveu model: $i\\partial_t\\psi=-i\\alpha\\partial_x\\psi+m\\beta\\psi-f(\\psi^\\ast\\beta\\psi)\\beta\\psi$, $\\psi(x,t)\\in\\C^2$, $x\\in\\R$, $f\\in C^\\infty(\\R)$, $m>0$, where $\\alpha$, $\\beta$ are $2\\times 2$ hermitian matrices which satisfy $\\alpha^2=\\beta^2=1$, $\\alpha\\beta+\\beta\\alpha=0$. We study the spectral stability of solitary wave solutions $\\phi_\\omega(x)e^{-i\\omega t}$. More precisely, we study the presence of point eigenvalues in the spectra of linearizations at solitary waves of arbitrarily small amplitude, in the limit $\\omega\\to m$. We prove that if $f(s)=s^k+O(s^{k+1})$, $k\\in\\N$, with $k\\ge 3$, then one positive and one negative eigenvalue are present in the spectrum of linearizations at all solitary waves with $\\omega$ sufficiently close to $m$. This shows that all solitary waves of sufficiently small amplitude are linearly unstable. The approach is based on applying the Rayleigh-Schr\\\"odinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov-Kolokolov stability criterion. Let us mention a similar independent result [Guan-Gustafson] on linear instability for the nonlinear Dirac equation in three dimensions, with cubic nonlinearity (this result is also in formal agreement with the Vakhitov-Kolokolov stability criterion).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-13T23:28:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35C08",
      "35P99",
      "35Q41",
      "37K40",
      "37K45",
      "81Q05"
    ],
    "keywords": [
      "nonlinear dirac equation",
      "higher order nonlinearity",
      "linear instability",
      "vakhitov-kolokolov stability criterion",
      "formal agreement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.3859C"
    }
  }
}