{
  "id": "2104.07301",
  "version": "v1",
  "published": "2021-04-15T08:22:51.000Z",
  "updated": "2021-04-15T08:22:51.000Z",
  "title": "Long time asymptotics for the focusing nonlinear Schrödinger equation in the solitonic region with the presence of high-order discrete spectrum",
  "authors": [
    "Zhaoyu Wang",
    "Meisen Chen",
    "Engui Fan"
  ],
  "comment": "44 pages",
  "categories": [
    "math.AP",
    "nlin.SI"
  ],
  "abstract": "In this paper, we use the $\\bar{\\partial}$ steepest descent method to study the initial value problem for focusing nonlinear Schr\\\"odinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we shall characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; further we formulate an appropriate enlarged RH problem; after a series of deformations, the RH problem is transformed into a solvable model. Finally, we obtain the asymptotic expansion of the solution of the fNLS equation in any fixed space-time cone: %as $t \\to \\infty$, \\begin{equation*} \\mathcal{S}(x_1,x_2,v_1,v_2):=\\left\\lbrace (x,t)\\in \\mathbb{R}^2: x=x_0+vt, \\ x_0\\in[x_1,x_2]\\text{, }v\\in[v_1,v_2]\\right\\rbrace. \\end{equation*} Observing the result indicates that the solution of fNLS equation in this case satisfies the soliton resolution conjecture. The leading order term of this solution includes a high-order pole-soliton whose parameters are affected by soliton-soliton interactions through the cone and soliton-radiation interactions on continuous spectrum. The error term of this result is up to $\\mathcal{O}(t^{-3/4})$ which comes from the corresponding $\\bar{\\partial}$ equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-15T08:22:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "focusing nonlinear schrödinger equation",
      "high-order discrete spectrum",
      "long time asymptotics",
      "solitonic region",
      "non-generic weighted sobolev initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}