{
  "id": "2402.11377",
  "version": "v1",
  "published": "2024-02-17T20:51:17.000Z",
  "updated": "2024-02-17T20:51:17.000Z",
  "title": "Reducibility of Klein-Gordon equations with maximal order perturbations",
  "authors": [
    "Massimiliano Berti",
    "Roberto Feola",
    "Michela Procesi",
    "Shulamit Terracina"
  ],
  "comment": "79 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation $\\psi_{tt}-\\psi_{xx}+\\mathtt{m}\\psi+Q(\\omega t)\\psi=0 $ where $ Q(\\omega t) := a^{(2)}(\\omega t, x) \\partial_{xx} + a^{(1)}(\\omega t, x)\\partial_x + a^{(0)}(\\omega t, x) $ is a differential operator of order $ 2 $, parity preserving and reversible, are almost periodic in time and uniformly bounded for all times, provided that the coefficients $ a^{(2) }, a^{(1) }, a^{(0) } $ are small enough and the forcing frequency $\\omega\\in {\\mathbb R}^{\\nu}$ belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation $ Q (\\omega t) $ of the second order differential operator $ a^{(2)}(\\omega t, x)\\partial_{xx} $. In suitable coordinates the Klein-Gordon equation is the composition of two backward/forward quasi-periodic in time perturbed transport equations with non-constant coefficients, up to lower order pseudo-differential remainders. A key idea is to straighten this first order pseudo-differential operator with bi-characteristics through a novel quantitative Egorov analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-17T20:51:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37K55",
      "35L05"
    ],
    "keywords": [
      "maximal order perturbations",
      "forced linear klein-gordon equation",
      "first order pseudo-differential operator",
      "lower order pseudo-differential remainders",
      "diagonal constant coefficient system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 79,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}