An improved uniform error bound at $O\left(h^m+\varepsilon^2 \tau^2\right)$ is established in $H^{\alpha/2}$-norm for the long-time dynamics of the nonlinear space fractional Klein-Gordon equation (NSFKGE). A second-order exponential wave integrator (EWI) method is used to semi-discretize NSFKGE in time and the Fourier spectral method in space is applied to derive the full-discretization scheme. Regularity compensation oscillation (RCO) technique is employed to prove the improved uniform error bounds at $O\left(\varepsilon^2 \tau^2\right)$ in temporal semi-discretization and $O\left(h^m+\varepsilon^2 \tau^2\right)$ in full-discretization up to the long-time $T_{\varepsilon}=T / \varepsilon^2$ ($T>0$ fixed), respectively. Complex NSFKGE and oscillatory complex NSFKGE with nonlinear terms of general power exponents are also discussed. Finally, the correctness of the theoretical analysis and the effectiveness of the method are verified by numerical examples.

翻译：本文针对非线性空间分数阶Klein-Gordon方程（NSFKGE）的长时间动力学，在$H^{\alpha/2}$范数下建立了改进的一致误差界$O\left(h^m+\varepsilon^2 \tau^2\right)$。时间上采用二阶指数波积分法进行半离散，空间上采用傅里叶谱方法推导全离散格式。利用正则补偿振荡技术，分别证明了在半离散格式下改进的一致误差界为$O\left(\varepsilon^2 \tau^2\right)$，在全离散格式下长时间尺度$T_{\varepsilon}=T / \varepsilon^2$（$T>0$固定）内为$O\left(h^m+\varepsilon^2 \tau^2\right)$。此外，还讨论了具有一般幂指数非线性项的复NSFKGE和振荡复NSFKGE。最后，通过数值算例验证了理论分析的正确性和方法的有效性。