Chapter 4 discussed the evolution of infinitesimal perturbations of a uniform state into saturated, stationary, spatially periodic solutions. By restricting attention to such solutions, we were able to study the effects of the nonlinearities, using analytical methods near threshold and numerical methods further from threshold. However, most realistic geometries do not permit spatially periodic solutions since these solutions are usually not compatible with the boundary conditions at the lateral walls. Even if periodic solutions are consistent with some finite domain, they do not exhaust all the possible patterns. As we have seen in Section 4.4, typically patterns have the ideal form (stripes, hexagons, etc.) only over small regions and these ideal forms are distorted over long length scales or disrupted in localized regions by defects. In addition, the distortions and defects are often time-dependent.

In this chapter, we introduce the amplitude equation formalism which provides a powerful and broadly useful method to study spatial and temporal distortions of ideal patterns. The formalism represents a substantial conceptual and technical simplification in that, near onset and for slowly varying distortions of periodic patterns, the evolution of the many fields u(x, t) that describe some physical system (e.g. temperature, velocity, and concentration fields) can be described quantitatively in terms of the evolution of a single scalar complex-valued field A(x, t) called the amplitude. The evolution equation for the amplitude is called the amplitude equation and is typically a partial differential equation (pde). Amplitude equations capture three basic ingredients of pattern formation: the growth of a perturbation about the spatially uniform state, the saturation of the growth by nonlinearity, and what we will loosely call dispersion, namely the effect of spatial distortions.