# Validity of amplitude equations for non-local non-linearities

## TL;DR

This paper investigates the validity of amplitude equations for PDEs with non-local nonlinearities, specifically quadratic and cubic convolution types, using the Swift-Hohenberg equation as a benchmark, and derives a Ginzburg-Landau PDE with kernel-dependent coefficients.

## Contribution

It extends the amplitude equation framework to non-local nonlinearities, providing a rigorous derivation and analysis for convolution-type interactions.

## Key findings

- Amplitude equations remain valid for non-local nonlinearities.
- Derived explicit formulas for Ginzburg-Landau coefficients from kernels.
- Established a proof based on Fourier mode separation and kernel bounds.

## Abstract

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.