{
  "id": "1904.02540",
  "version": "v1",
  "published": "2019-04-04T13:26:29.000Z",
  "updated": "2019-04-04T13:26:29.000Z",
  "title": "Orbital Stability of Standing Waves for BNLS]{Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions",
  "authors": [
    "Tingjian Luo",
    "Shijun Zheng",
    "Shihui Zhu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the standing wave solutions of the bi-harmonic nonlinear Schr\\\"{o}dinger equation with the Laplacian term (BNLS). By taking into account the role of second-order dispersion term in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity, we prove that in the mass-subcritical regime $p\\in (1,1+\\frac{8}{d})$, there exist orbitally stable standing waves for BNLS, when $\\mu\\geq 0$, or $-\\lambda_0\\le\\mu<0$, for some $\\lambda_0>0$. Moreover, we prove that in the mass-critical case $p=1+\\frac{8}{d}$, the BNLS is orbital stable when $-\\lambda_1\\le\\mu<0$, for some $\\lambda_1>0$, and the initial data is below the ground state for the bihamonic operator. This shows that the sign of the second-order dispersion has crucial effect on the existence of orbitally stable standing waves for the BNLS with the mixed dispersions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-04T13:26:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "orbital stability",
      "fourth-order nonlinear schrödinger equation",
      "mixed dispersions",
      "orbitally stable standing waves",
      "second-order dispersion term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}