{
  "id": "0904.0834",
  "version": "v1",
  "published": "2009-04-06T02:29:16.000Z",
  "updated": "2009-04-06T02:29:16.000Z",
  "title": "Solitary waves for the Hartree equation with a slowly varying potential",
  "authors": [
    "Kiril Datchev",
    "Ivan Ventura"
  ],
  "comment": "28 pages",
  "journal": "Pacific Journal of Mathematics, Vol. 248, No. 1, pp. 63-90, 2010",
  "doi": "10.2140/pjm.2010.248.63",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the Hartree equation with a slowly varying smooth potential, $V(x) = W(hx)$, and with an initial condition which is $\\epsilon \\le \\sqrt h$ away in $H^1$ from a soliton. We show that up to time $|\\log h|/h$ and errors of size $\\epsilon + h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer-Zworski, who prove a similar theorem for the Gross-Pitaevskii equation, and on spectral estimates for the linearized Hartree operator recently obtained by Lenzmann. We also provide an extension of the result of Holmer-Zworski to more general inital conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-06T02:29:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hartree equation",
      "slowly varying potential",
      "solitary waves",
      "general inital conditions",
      "slowly varying smooth potential"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.0834D"
    }
  }
}