{
  "id": "1207.1989",
  "version": "v2",
  "published": "2012-07-09T09:07:54.000Z",
  "updated": "2013-01-22T10:30:06.000Z",
  "title": "Bifurcation in a multi-component system of nonlinear Schrödinger equations",
  "authors": [
    "Thomas Bartsch"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the system -\\Delta u_j + a(x)u_j = \\mu_j u_j^3 + \\be\\sum_{k\\ne j}u_k^2u_j, u_j>0, \\qquad j=1,...,n, on a possibly unbounded domain $\\Om\\subset\\R^N$, $N\\le3$, with Dirichlet boundary conditions. The system appears in nonlinear optics and in the analysis of mixtures of Bose-Einstein condensates. We consider the self-focussing (attractive self-interaction) case $\\mu_1,...,\\mu_n > 0$ and take $\\be\\in\\R$ as bifurcation parameter. There exists a branch of positive solutions with $u_j/u_k$ being constant for all $j,k\\in{1,...,n}$. The main results are concerned with the bifurcation of solutions from this branch. Using a hidden symmetry we are able to prove global bifurcation even when the linearization has even-dimensional kernel (which is always the case when $n>1$ is odd).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-22T10:30:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B05",
      "35B32",
      "35J50",
      "35J55",
      "58C40",
      "58E07"
    ],
    "keywords": [
      "nonlinear schrödinger equations",
      "multi-component system",
      "dirichlet boundary conditions",
      "nonlinear optics",
      "even-dimensional kernel"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.1989B"
    }
  }
}