In this paper, a Fourier spectral method with an adaptive time step strategy is proposed to solve the fractional nonlinear Schrödinger (FNLS) equation with periodic initial value problem. First, we prove the conservation law of the mass and the energy for the semi‐discrete Fourier spectral scheme. Second, the error estimation of the semi‐discrete scheme is given in the relevant fractional Sobolev space. Then, an adaptive time‐step strategy is designed to reduce central processing unit (CPU) time. Finally, the numerical experiments for the one‐, two‐ and three‐dimensional FNLSs, show that the adaptive strategy, compared to the constant time step, can reduce the CPU‐time by almost half.

中文翻译：

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带有非线性非线性薛定ding方程的自适应时间策略的傅立叶谱方法

本文提出了一种具有自适应时步策略的傅立叶谱方法，以求解带有周期初值问题的分数阶非线性薛定ding（FNLS）方程。首先，我们证明了半离散傅立叶频谱方案的质量守恒律和能量守恒律。其次，在相关的分数Sobolev空间中给出了半离散方案的误差估计。然后，设计了自适应时间步长策略以减少中央处理器（CPU）的时间。最后，针对一维，二维和三维FNLS的数值实验表明，与恒定时间步长相比，自适应策略可以将CPU时间减少近一半。