A conservative numerical method for the fractional nonlinear Schröinger equation in two dimensions

  • Rongpei ZhangEmail author
  • Yong-Tao Zhang
  • Zhen Wang
  • Bo Chen
  • Yi Zhang


This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrödinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrödinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.


fractional nonlinear Schrödinger equation weighted and shifted Grünwald-Letnikov difference compact integration factor method conservation 


65N06 35R11 


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This work was supported by National Natural Science Foundation of China (Grant Nos. 61573008 and 61703290), the Foundation of LCP (Grant No. 6142A0502020717) and National Science Foundation of USA (Grant No. DMS-1620108).


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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Rongpei Zhang
    • 1
    Email author
  • Yong-Tao Zhang
    • 2
  • Zhen Wang
    • 3
  • Bo Chen
    • 4
  • Yi Zhang
    • 1
  1. 1.School of Mathematics and Systematic SciencesShenyang Normal UniversityShenyangChina
  2. 2.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameUSA
  3. 3.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  4. 4.College of Mathematics and StatisticsShenzhen UniversityShenzhenChina

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