{
  "id": "1810.04944",
  "version": "v1",
  "published": "2018-10-11T10:34:05.000Z",
  "updated": "2018-10-11T10:34:05.000Z",
  "title": "Coupled Mode Equations and Gap Solitons in Higher Dimensions",
  "authors": [
    "Tomas Dohnal",
    "Lisa Wahlers"
  ],
  "comment": "27 pages, 6 figures",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "nlin.PS"
  ],
  "abstract": "We study waves-packets in nonlinear periodic media in arbitrary ($d$) spatial dimension, modeled by the cubic Gross-Pitaevskii equation. In the asymptotic setting of small and broad waves-packets with $N\\in \\mathbb{N}$ carrier Bloch waves the effective equations for the envelopes are first order coupled mode equations (CMEs). We provide a rigorous justification of the effective equations. The estimate of the asymptotic error is carried out in an $L^1$-norm in the Bloch variables. This translates to a supremum norm estimate in the physical variables. In order to investigate the existence of gap solitons of the $d$-dimensional CMEs, we discuss spectral gaps of the CMEs. For $N=4$ and $d=2$ a family of time harmonic gap solitons is constructed formally asymptotically and numerically. Moving gap solitons have not been found for $d>1$ and for the considered values of $N$ due to the absence of a spectral gap in the standard moving frame variables.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-11T10:34:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35Q60",
      "35L71",
      "41A60"
    ],
    "keywords": [
      "higher dimensions",
      "time harmonic gap solitons",
      "first order coupled mode equations",
      "spectral gap",
      "standard moving frame variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}