{
  "id": "1908.11629",
  "version": "v1",
  "published": "2019-08-30T10:09:45.000Z",
  "updated": "2019-08-30T10:09:45.000Z",
  "title": "Normalized solutions for a coupled Schrödinger system",
  "authors": [
    "Thomas Bartsch",
    "Xuexiu Zhong",
    "Wenming Zou"
  ],
  "comment": "25 pages, 1 figure",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the present paper, we prove the existence of solutions $(\\lambda_1,\\lambda_2,u,v)\\in\\mathbb{R}^2\\times H^1(\\mathbb{R}^3,\\mathbb{R}^2)$ to systems of coupled Schr\\\"odinger equations $$ \\begin{cases} -\\Delta u+\\lambda_1u=\\mu_1 u^3+\\beta uv^2\\quad &\\hbox{in}\\;\\mathbb{R}^3\\\\ -\\Delta v+\\lambda_2v=\\mu_2 v^3+\\beta u^2v\\quad&\\hbox{in}\\;\\mathbb{R}^3\\\\ u,v>0&\\hbox{in}\\;\\mathbb{R}^3 \\end{cases} $$ satisfying the normalization constraint $ \\displaystyle\\int_{\\mathbb{R}^3}u^2=a^2\\quad\\hbox{and}\\;\\int_{\\mathbb{R}^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $\\mu_1,\\mu_2,\\beta>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the fixed frequency case. And when the masses are prescribed, the standard approach to this problem is variational with $\\lambda_1,\\lambda_2$ appearing as Lagrange multipliers. Here we present a new approach based on bifurcation theory and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\\beta$ in a large range. We also give a result about the nonexistence of positive solutions. From which one can see that our existence theorem is almost the best. Especially, if $\\mu_1=\\mu_2$ we prove that normalized solutions exist for all $\\beta>0$ and all $a,b>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-30T10:09:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35Q51",
      "35B09",
      "35B32",
      "35B40"
    ],
    "keywords": [
      "normalized solutions",
      "coupled schrödinger system",
      "fixed frequency case",
      "large range",
      "continuation method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}