Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical and numerical approaches. The purpose of this work is to study the effects of the spectral fractional Laplacian on the bifurcation structure of reaction–diffusion systems on bounded domains. In order to do this we use advanced numerical continuation techniques to compute the solution branches. Since current available continuation packages only support systems involving the standard Laplacian, we first extend the pde2path software to treat fractional PDEs (in the spectral definition). The new capabilities are then applied to the study of the Allen–Cahn equation, the Swift–Hohenberg equation and the Schnakenberg system (in which the standard Laplacian is replaced by the spectral fractional Laplacian). In particular, we investigate the changes in snaking bifurcation diagrams and in the spatial structure of non-trivial steady states upon variation of the order of the fractional Laplacian. Our results show that the fractional order induces significant qualitative and quantitative changes in the overall bifurcation structures, of which some are shared by the three systems. This contributes to a better understanding of the effects of fractional diffusion in generic reaction–diffusion systems.