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Nonlinear Dynamics

, Volume 98, Issue 2, pp 985–995 | Cite as

One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential

  • Liangwei Zeng
  • Jianhua ZengEmail author
Original paper

Abstract

Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear terms whose strengths are of opposite signs (the coefficients in front of the nonlinearities), exist in various physical media (in particular, in optical and matter-wave media). A benign competition between self-focusing cubic and self-defocusing quintic nonlinear nonlinearities (known as cubic–quintic model) plays an important role in creating and stabilizing the self-trapping of D-dimensional localized structures, in the contexts of standard nonlinear Schrödinger equation. We incorporate an external periodic potential (linear lattice) into this model and extend it to the space-fractional scenario that begins to surface in very recent years—the nonlinear fractional Schrödinger equation (NLFSE), therefore obtaining the cubic–quintic or the purely quintic NLFSE, and investigate the propagation and stability properties of self-trapped modes therein. Two types of one-dimensional localized gap modes are found, including the fundamental and dipole-mode gap solitons. Employing the techniques based on the linear-stability analysis and direct numerical simulations, we get the stability regions of all the localized modes; and particularly, the anti-Vakhitov–Kolokolov criterion applies for the stable portions of soliton families generated in the frameworks of quintic-only nonlinearity and competing cubic–quintic nonlinear terms.

Keywords

Fractional calculus Cubic–quintic nonlinearity Nonlinear Schrödinger equation Gap solitons 

Notes

Acknowledgements

This work was supported, in part, by the Natural Science Foundation of China (Project Nos. 61690222, 61690224), and by the Youth Innovation Promotion Association of the Chinese Academy of Sciences (Project No. 2016357).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. 1.State Key Laboratory of Transient Optics and PhotonicsXi’an Institute of Optics and Precision Mechanics of Chinese Academy of SciencesXi’anChina
  2. 2.University of Chinese Academy of SciencesBeijingChina

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