The amplitude equation for the space-fractional Swift–Hohenberg equation

Abstract

Non-local reaction–diffusion partial differential equations (PDEs) involving the fractional Laplacian have arisen in a wide variety of applications. One common tool to analyze the dynamics of classical local PDEs very close to instability is to derive local amplitude/modulation multiscale approximations, which provide local normal forms classifying the onset of 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, i.e., to rigorously prove a multiscale decomposition between the leading critical modes and the higher-order remainder terms.