Validity of amplitude equations for nonlocal nonlinearities

Amplitude equations are used to describe the onset of instability in wide classes of partial differential equations (PDEs). One goal of the field is to determine simple universal/generic PDEs, to which many other classes of equations can be reduced, at least on a sufficiently long approximating time scale. In this work, we study the case when the reaction terms are nonlocal. In particular, we consider quadratic and cubic convolution-type nonlinearities. As a benchmark problem, we use the Swift-Hohenberg equation. The resulting amplitude equation is a Ginzburg-Landau PDE, where the coefficients can be calculated from the kernels. Our proof relies on separating critical and noncritical modes in Fourier space in combination with suitable kernel bounds.