  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.
