The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

Peiyao Wang, Shangwen Peng, Yihao Cao and Rongpei Zhang ()
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Peiyao Wang: School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China
Shangwen Peng: School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China
Yihao Cao: School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China
Rongpei Zhang: School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China

Mathematics, 2024, vol. 12, issue 7, 1-14

Abstract: This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

Keywords: nonlinear Schrödinger equation; fractional Laplace operator; optical solitons; conservation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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