The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.

非线性空间分数问题通常允许多个平稳解，这可能比相应的整数阶问题复杂得多。在本文中，我们系统地计算了一维和二维矩形域上非线性常数/变阶空间分数问题的解决方案。提出了一种快速逼近算法，通过逼近变指数傅里叶模式来处理变阶谱分数拉普拉斯算子，然后结合鞍动力学构造变阶空间分数相场模型的解图。通过数值实验验证了快速逼近算法的准确性和效率，阐明了空间分数相场模型稳态解的基本特征。此外，