  Non-local reaction-diffusion partial differential equations (PDEs) involving
the fractional Laplacian have arisen in a wide variety of applications. One
common tool to analyse the dynamics of classical local PDEs near instability is
to derive local amplitude/modulation approximations, which provide local normal
forms classifying a wide variety of pattern-formation phenomena. In this work,
we study amplitude equations for the space-fractional Swift-Hohenberg equation.
The Swift-Hohenberg equation is a basic model problem motivated by pattern
formation in fluid dynamics and has served as one of the main PDEs to develop
general techniques to derive amplitude equations. We prove that there exists
near the first bifurcation point an approximation by a (real) Ginzburg-Landau
equation. Interestingly, this Ginzburg-Landau equation is a local PDE, which
provides a rigorous justification of the physical conjecture that suitably
localized unstable modes can out-compete superdiffusion and re-localize a PDE
near instability. Our main technical contributions are to provide a suitable
function space setting for the approximation problem, and to then bound the
residual between the original PDE and its amplitude equation.
