In Chapter 7 we allowed patterns to deviate slightly from a regular lattice by permitting modulations on long scales. However, in a large-aspect-ratio system that can accommodate a large number of pattern wavelengths in all directions, the size and orientation of the pattern will typically change slowly in space and time. In spiral defect chaos, for example, you tend to see patches of rolls that look quite regular locally, but in fact are curved with a large radius of curvature (Figure 10.17). Fingerprints, though stationary, also look like stripes that vary slowly in orientation over a large domain. The ridges in fingerprints are believed to form through the buckling of the lower layer of the skin; recently Kuecken (2004) has derived roll-hexagon amplitude equations from a buckling model of fingerprint formation, suggesting that the analysis of fingerprints as a pattern-forming system may be valid.

Obviously we can't describe patterns in a large-aspect-ratio system by assuming that they lie almost on a lattice, since they clearly don't. However, far from onset in the fully nonlinear regime, we can use the slowness with which the patterns evolve in time and space to develop an asymptotic description of them. The full nonlinearity is a requirement of the theory, so we will lose the small parameter measuring the distance from onset that we used previously to derive amplitude equations, but the slow rates of change will give us a new small parameter to work with.

The theory presented in this chapter was originally developed by Cross and Newell (1984) and later expanded by Passot and Newell (1994). Here we follow their treatment of the problem quite closely.