The aim of this paper is to establish improved uniform error bounds under \(\varvec{H^{\alpha /2}}\)-norm \(\varvec{(1<\alpha \le 2)}\) for the long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation (NSFSGE) by a Lawson-type exponential integrator Fourier pseudo-spectral (LEI-FP) method. Firstly, a Lawson-type exponential integrator method is used to discretize the time direction. Then, the Fourier pseudo-spectral method is applied to discretize the space direction. We rigorously prove that the equation is energy conservation in a continuous state. Regularity compensation oscillation (RCO) technique is employed to strictly prove the improved uniform error bounds at \(\varvec{O\left( \varepsilon ^2 \tau \right) }\) in temporal semi-discretization and \(\varvec{O\left( h^m+\varepsilon ^2 \tau \right) }\) in full-discretization up to the long-time \(\varvec{T_{\varepsilon }=T / \varepsilon ^2}\) (\(\varvec{T>0}\) fixed), respectively. To obtain the convergence order \(\varvec{h^{m}}\) in space, we only need to directly prove it instead of proving that the numerical solution is \(\varvec{H^{m+\alpha /2}}\)-norm bounded as before. Complex NSFSGE and oscillatory NSFSGE are also discussed. This is the novel work to construct the improved uniform error bounds for the long-time dynamics of the high-dimensional nonlinear space fractional Klein-Gordon equation with non-polynomial nonlinearity. Finally, numerical examples in two-dimension and three-dimension are provided to confirm the improved error bounds, and we find drastically different evolving patterns between NSFSGE and the classical sine-Gordon equation.

本文的目的是在\(\varvec{H^{\alpha /2}}\) -norm \(\varvec{(1<\alpha \le 2)}\)下建立改进的统一误差界采用 Lawson 型指数积分傅立叶伪谱 (LEI-FP) 方法对高维非线性空间分数正弦戈登方程 (NSFSGE) 进行长期动力学分析。首先，采用Lawson型指数积分器方法对时间方向进行离散化。然后，应用傅立叶伪谱方法对空间方向进行离散化。我们严格证明该方程是连续状态下的能量守恒。采用正则性补偿振荡（RCO）技术严格证明时间半离散化中\(\varvec{O\left( \varepsilon ^2 \tau \right) }\)和\(\varvec{ O\left( h^m+\varepsilon ^2 \tau \right) }\)完全离散化直至长时间\(\varvec{T_{\varepsilon }=T / \varepsilon ^2}\) ( \(\varvec{T>0}\)固定) 分别。要获得空间上的收敛阶数\(\varvec{h^{m}}\)，我们只需直接证明即可，而不需要证明数值解为\(\varvec{H^{m+\alpha /2} }\) -范数与以前一样有界。还讨论了复杂 NSFSGE 和振荡 NSFSGE。这是为具有非多项式非线性的高维非线性空间分数式 Klein-Gordon 方程的长期动力学构造改进的均匀误差界的新颖工作。最后，提供了二维和三维的数值例子来确认改进的误差范围，我们发现 NSFSGE 和经典的正弦戈登方程之间的演化模式截然不同。