In this chapter, we discuss model evolution equations that serve as a bridge between the previous chapter on the qualitative properties of nonlinear saturated states and later chapters on the amplitude equation and phase diffusion equation, which provide a way to understand many quantitative details of pattern formation. Model equations are a natural next step after the previous chapter because they demonstrate a nontrivial insight, that many details of experimental pattern formation can be understood without having to work with the quantitatively accurate but often difficult evolution equations that describe pattern formation in say a liquid crystal or some reaction–diffusion system. Analytical and numerical calculations show that if a simplified model contains certain symmetries (say rotational, translational, and inversion), has structure at a preferred length scale (generated say by a type-I instability), and has a nonlinearity that saturates exponentially growing modes, the model is often able to reproduce qualitative features, and in some cases quantitative details, observed in experiments.

The same insight that mathematically simplified models can have a rich pattern formation is useful for later chapters because model equations provide a more efficient way to carry out and test theoretical formalisms than would be the case for quantitatively accurate evolution equations. We have already used the Swift–Hohenberg model in this way, for example to calculate how the growth rate σq of an unstable mode depends on the perturbation wave number q (Eq. (2.11)), or to calculate approximate stationary nonlinear stripe solutions near onset (Eqs. (4.21) and (4.24)). Model equations can also usually be studied numerically more thoroughly than the fully quantitative equations.