  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 non-local.
In particular, we consider quadratic and cubic convolution-type
non-linearities. 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 non-critical modes in Fourier space in combination with suitable
kernel bounds.
