{
  "id": "2203.03500",
  "version": "v1",
  "published": "2022-03-07T16:26:02.000Z",
  "updated": "2022-03-07T16:26:02.000Z",
  "title": "Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators",
  "authors": [
    "Jean-Baptiste Casteras",
    "Juraj Foldes",
    "Gennady Uraltsev"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "In this paper, we study the local well-posedness of the cubic Schr\\\" odinger equation: $$ (i\\partial_t - P) u = \\pm |u|^2 u\\quad \\textrm{ on }\\ I\\times \\R^d ,$$ with randomized initial data, and $P$ being an operator of degree $s \\geq 2$. Using careful estimates in anisotropic spaces, we improve and extend known results for the standard Schr\\\"odinger equation (that is, $P$ being Laplacian) to any dimension under natural assumptions on $P$, whose Fourier symbol might be sign changing. Quite interestingly, we also exhibit the existence of a new regime depending on $s$ and $d$, which was not present for the Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-07T16:26:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35G20",
      "35A01",
      "35R60"
    ],
    "keywords": [
      "cubic nonlinear schrodinger equation",
      "sure local well-posedness",
      "higher order operators",
      "randomized initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}