{
  "id": "2112.12119",
  "version": "v2",
  "published": "2021-12-22T18:31:48.000Z",
  "updated": "2023-09-08T12:31:00.000Z",
  "title": "On the Well-posedness and Stability of Cubic and Quintic Nonlinear Schrödinger Systems on ${\\mathbb T}^3$",
  "authors": [
    "Thomas Chen",
    "Amie Urban"
  ],
  "comment": "AMS Latex, 32 pages",
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "In this paper, we study cubic and quintic nonlinear Schr\\\"odinger systems on 3-dimensional tori, with initial data in an adapted Hilbert space $H^s_{\\underline{\\lambda}},$ and all of our results hold on rational and irrational rectangular, flat tori. In the cubic and quintic case, we prove local well-posedness for both focusing and defocusing systems. We show that local solutions of the defocusing cubic system with initial data in $H^1_{\\underline{\\lambda}}$ can be extended for all time. Additionally, we prove that global well-posedness holds in the quintic system, focusing or defocusing, for initial data with sufficiently small $H^1_{\\underline{\\lambda}}$ norm. Finally, we use the energy-Casimir method to prove the existence and uniqueness, and nonlinear stability of a class of stationary states of the defocusing cubic and quintic nonlinear Schr\\\"odinger systems.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-09-08T12:31:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82D10",
      "82C10"
    ],
    "keywords": [
      "quintic nonlinear schrödinger systems",
      "initial data",
      "global well-posedness holds",
      "defocusing cubic",
      "nonlinear stability"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}