{
  "id": "2407.00300",
  "version": "v1",
  "published": "2024-06-29T03:45:37.000Z",
  "updated": "2024-06-29T03:45:37.000Z",
  "title": "On the near soliton dynamics for the 2D cubic Zakharov-Kuznetsov equations",
  "authors": [
    "Gong Chen",
    "Yang Lan",
    "Xu Yuan"
  ],
  "comment": "65 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article, we consider the Cauchy problem for the cubic (mass-critical) Zakharov-Kuznetsov equations in dimension two: $$\\partial_t u+\\partial_{x_1}(\\Delta u+u^3)=0,\\quad (t,x)\\in [0,\\infty)\\times \\mathbb{R}^{2}.$$ For initial data in $H^1$ close to the soliton with a suitable space-decay property, we fully describe the asymptotic behavior of the corresponding solution. More precisely, for such initial data, we show that only three possible behaviors can occur: 1) The solution leaves a tube near soliton in finite time; 2) the solution blows up in finite time; 3) the solution is global and locally converges to a soliton. In addition, we show that for initial data near a soliton with non-positive energy and above the threshold mass, the corresponding solution will blow up as described in Case 2. Our proof is inspired by the techniques developed for mass-critical generalized Korteweg-de Vries equation (gKdV) equation in a similar context by Martel-Merle-Rapha\\\"el. More precisely, our proof relies on refined modulation estimates and a modified energy-virial Lyapunov functional. The primary challenge in our problem is the lack of coercivity of the Schr\\\"odinger operator which appears in the virial-type estimate. To overcome the difficulty, we apply a transform, which was first introduced in Kenig-Martel [13], to perform the virial computations after converting the original problem to the adjoint one. Th coercivity of the Schr\\\"odinger operator in the adjoint problem has been numerically verified by Farah-Holmer-Roudenko-Yang [9].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-29T03:45:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "2d cubic zakharov-kuznetsov equations",
      "soliton dynamics",
      "initial data",
      "finite time",
      "mass-critical generalized korteweg-de vries equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}