Orbital stability of periodic standing waves for the cubic fractional nonlinear Schrödinger equation


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Bittencourt Moraes G. E., Borluk H., de Loreno G., Muslu G. M., Natali F.

Journal of Differential Equations, vol.341, pp.263-291, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 341
  • Publication Date: 2022
  • Doi Number: 10.1016/j.jde.2022.09.015
  • Journal Name: Journal of Differential Equations
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.263-291
  • Keywords: Fractional Schr?dinger equation, Existence and uniqueness of minimizers, Small-amplitude periodic waves, Orbital stability
  • Istanbul Medipol University Affiliated: No

Abstract

In this paper, the existence and orbital stability of the periodic standing wave solutions for the nonlinear fractional Schrödinger (fNLS) equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and it is showed that the corresponding real solution is always positive. The orbital stability is proved by combining some tools regarding the oscillation theorem for fractional Hill operators and the Vakhitov-Kolokolov condition, well known for Schrödinger equations. We then perform a numerical approach to generate the periodic standing wave solutions of the fNLS equation by using the Petviashvili's iteration method. We also investigate the Vakhitov-Kolokolov condition numerically which cannot be obtained analytically for some values of the order of the fractional derivative.