In this paper, we derive the improved uniform error bounds for the long-time dynamics of the $d$-dimensional $(d=2,3)$ nonlinear space fractional sine-Gordon equation (NSFSGE). The nonlinearity strength of the NSFSGE is characterized by $\varepsilon^2$ where $0<\varepsilon \le 1$ is a dimensionless parameter. The second-order time-splitting method is applied to the temporal discretization and the Fourier pseudo-spectral method is used for the spatial discretization. To obtain the explicit relation between the numerical errors and the parameter $\varepsilon$, we introduce the regularity compensation oscillation technique to the convergence analysis of fractional models. Then we establish the improved uniform error bounds $O\left(\varepsilon^2 \tau^2\right)$ for the semi-discretization scheme and $O\left(h^m+\varepsilon^2 \tau^2\right)$ for the full-discretization scheme up to the long time at $O(1/\varepsilon^2)$. Further, we extend the time-splitting Fourier pseudo-spectral method to the complex NSFSGE as well as the oscillatory complex NSFSGE, and the improved uniform error bounds for them are also given. Finally, extensive numerical examples in two-dimension or three-dimension are provided to support the theoretical analysis. The differences in dynamic behaviors between the fractional sine-Gordon equation and classical sine-Gordon equation are also discussed.

翻译：本文推导了$d$维（$d=2,3$）非线性空间分数阶正弦-戈登方程（NSFSGE）长时间动力学的改进均匀误差界。该方程的弱非线性强度由无量纲参数$\varepsilon^2$表征，其中$0<\varepsilon\le 1$。时间离散采用二阶时间分裂方法，空间离散采用傅里叶伪谱方法。为建立数值误差与参数$\varepsilon$的显式关系，我们将正则补偿振荡技术引入分数阶模型的收敛性分析。进而建立了半离散格式在长时间$O(1/\varepsilon^2)$内的改进均匀误差界$O\left(\varepsilon^2 \tau^2\right)$，以及全离散格式的改进均匀误差界$O\left(h^m+\varepsilon^2 \tau^2\right)$。此外，我们将时间分裂傅里叶伪谱方法推广至复值NSFSGE及其振荡形式，并给出了相应的改进均匀误差界。最后，通过二维和三维的大量数值算例验证了理论分析，并讨论了分数阶与经典正弦-戈登方程动力学行为的差异。