Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation

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• 作者：Siwei Duo ^{sddy9@mst.edu" class="auth_mail" title="E-mail the corresponding author} ; Yanzhi Zhang ; ^{zhangyanz@mst.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional nonlinear Schrö ; dinger equation ; Fractional Laplacian ; Split-step method ; Crank&ndash ; Nicolson method ; Relaxation method
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：71
• 期：11
• 页码：2257-2271
• 全文大小：1463 K

文摘

We propose three Fourier spectral methods, i.e., the split-step Fourier spectral (SSFS), the Crank–Nicolson Fourier spectral (CNFS), and the relaxation Fourier spectral (ReFS) methods, for solving the fractional nonlinear Schrödinger (NLS) equation. All of them are mass conservative and time reversible, and they have the spectral order accuracy in space and the second-order accuracy in time. In addition, the CNFS and ReFS methods are energy conservative. The performance of these methods in simulating the plane wave and soliton dynamics is discussed. The SSFS method preserves the dispersion relation, and thus it is more accurate for studying the long-time behaviors of the plane wave solutions. Furthermore, our numerical simulations suggest that the SSFS method is better in solving the defocusing NLS, but the CNFS and ReFS methods are more effective for the focusing NLS.