In Chapter 2 we saw that a system with discrete eigenmodes can be reduced to its evolution on a centre manifold. By requiring patterns to be periodic with respect to a lattice in Chapter 5 we could distinguish between critical and decaying modes and apply the centre manifold theorem to extract amplitude equations for the critical modes. However, if the pattern is not exactly periodic our analysis must allow for the possibility that modes arbitrarily close to the critical modes in Fourier space contribute to the pattern. Then the distinction between stable and unstable modes becomes a little blurred: a mode with growth rate infinitesimally greater than zero will grow, but infinitely slowly, whereas a mode with growth rate infinitesimally less than zero will decay, but again infinitely slowly. In this case, we cannot perform a centre manifold reduction, since we cannot separate the growing and decaying modes well enough. Specifically, we cannot find an appropriate δ in equation (2.30) of Chapter 2. In cases such as these we must use an alternative method of analysis. This chapter describes how envelope equations can be used to describe the evolution of patterns that fit almost, but not exactly, onto a lattice.

Envelope equations for specific models

As explained in Chapter 5 pattern-forming systems can often be described adequately by a set of partial differential equations for a marker quantity, such as the density or temperature perturbation in a convecting fluid, together with appropriate boundary conditions. In this form, the problem is amenable to analysis using envelope equations. To explain the method, we will look at a specific example.