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. 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.

翻译：非线性空间折射问题往往允许多种固定式解决办法,这比相应的整数顺序问题复杂得多。在本文中,我们系统地计算非线性常数/可变顺序空间折射问题的解决方案面貌。开发了一个快速近似算法,通过接近可变分数分数模式处理可变分数谱分数拉帕西安,然后与马鞍动态结合,以构建可变-顺序空间折射场模型的解决方案面貌。进行了数字实验,以证实快速近似算法的准确性和效率,并阐明空间折射场模型的固定式解决方案的基本特征。此外,我们证明光谱分数分数拉帕西安问题解决方案面貌可以通过相应整数问题中不同的扩散系数来重新配置。