{
  "id": "2308.02297",
  "version": "v1",
  "published": "2023-08-04T13:02:30.000Z",
  "updated": "2023-08-04T13:02:30.000Z",
  "title": "Flat blow-up solutions for the complex Ginzburg Landau equation",
  "authors": [
    "Giao Ky Duong",
    "Nejla Nouaili",
    "Hatem Zaag"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the complex Ginzburg Landau equation $$ \\partial_t u = (1 + i \\beta ) \\Delta u + (1 + i \\delta) |u|^{p-1}u - \\alpha u \\text{ where } \\beta, \\delta, \\alpha \\in \\mathbb R. $$ The study aims to investigate the finite time blowup phenomenon. In particular, for fixed $ \\beta\\in \\mathbb R$, the existence of finite time blowup solutions for an arbitrary large $|\\delta|$ is still unknown. Especially, Popp, Stiller, Kuznetsov, and Kramer formally conjectured in 1998 that there is no blowup (collapse) in such a case. In this work, considered as a breakthrough, we give a counter example to this conjecture. We show the existence of blowup solutions in one dimension with $\\delta $ arbitrarily given and $\\beta =0$. The novelty is based on two main contributions: an investigation of a new blowup scaling (flat blowup regime) and a suitable modulation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-04T13:02:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K05",
      "35B40",
      "35K55",
      "35K57"
    ],
    "keywords": [
      "complex ginzburg landau equation",
      "flat blow-up solutions",
      "finite time blowup solutions",
      "finite time blowup phenomenon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}