{
  "id": "1209.1146",
  "version": "v3",
  "published": "2012-09-05T23:43:22.000Z",
  "updated": "2013-06-14T05:49:36.000Z",
  "title": "On linear instability of solitary waves for the nonlinear Dirac equation",
  "authors": [
    "Andrew Comech",
    "Meijiao Guan",
    "Stephen Gustafson"
  ],
  "comment": "17 pages. arXiv admin note: substantial text overlap with arXiv:1203.3859 (an earlier 1D version)",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.SP",
    "nlin.PS"
  ],
  "abstract": "We consider the nonlinear Dirac equation, also known as the Soler model: $i\\p\\sb t\\psi=-i\\alpha \\cdot \\nabla \\psi+m \\beta \\psi-f(\\psi\\sp\\ast \\beta \\psi) \\beta \\psi$, $\\psi(x,t)\\in\\mathbb{C}^{N}$, $x\\in\\mathbb{R}^n$, $n\\le 3$, $f\\in C\\sp 2(\\R)$, where $\\alpha_j$, $j = 1,...,n$, and $\\beta$ are $N \\times N$ Hermitian matrices which satisfy $\\alpha_j^2=\\beta^2=I_N$, $\\alpha_j \\beta+\\beta \\alpha_j=0$, $\\alpha_j \\alpha_k + \\alpha_k \\alpha_j =2 \\delta_{jk} I_N$. We study the spectral stability of solitary wave solutions $\\phi(x)e^{-i\\omega t}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $\\omega\\to m$, proving that if $k>2/n$, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with $\\omega$ sufficiently close to $m$, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh--Schroedinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov--Kolokolov stability criterion.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-06-14T05:49:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35C08",
      "35P99",
      "35Q41",
      "37K40",
      "37K45",
      "81Q05"
    ],
    "keywords": [
      "nonlinear dirac equation",
      "linear instability",
      "solitary wave solutions",
      "rayleigh-schroedinger perturbation theory",
      "vakhitov-kolokolov stability criterion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.1146C"
    }
  }
}