On a conservative Fourier spectral Galerkin method for cubic nonlinear Schrödinger equation with fractional Laplacian

Guang-an Zou, Bo Wang and Tony W.H. Sheu

Mathematics and Computers in Simulation (MATCOM), 2020, vol. 168, issue C, 122-134

Abstract: In this paper, a Crank–Nicolson Fourier spectral Galerkin method is proposed for solving the cubic fractional Schrödinger equation. Firstly, we discuss the mass and energy conservation laws for the nonlinear system and its corresponding fully discrete scheme. Secondly, the convergence with the spectral order accuracy in space and the second order of accuracy in time is exhibited. We perform one-dimensional calculation of the fractional derivative differential equation to verify our theoretical findings. Moreover, the proposed scheme is successfully applied to study two- and three-dimensional fractional quantum mechanics. Numerical results clearly exhibit that the fractional order can affect the shapes of soliton and rogue waves. The evolution of ground state solution can be clearly seen to be non-symmetrically configured when the fractional order becomes smaller.

Keywords: Fractional Schrödinger equation; Fourier spectral Galerkin method; Conservative laws; Convergence analysis; Numerical examples (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:168:y:2020:i:c:p:122-134

DOI: 10.1016/j.matcom.2019.08.006

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