Patterns such as hexagons and squares can also become unstable to phase and cross-pattern modes, while stripes in systems with additional symmetries, such as Galilean invariance, can undergo new types of instability, leading to drift, for example. This chapter looks at some of these new situations, starting with two examples of more complicated planforms – hexagons and quasipatterns – and some of their instabilities. After that we study drift instabilities where stationary or standing-wave patterns start to travel, and finally we look at the effect of Galilean invariance and conservation laws on the instabilities of stripes.

Instabilities of two-dimensional steady patterns

There are many possible extensions of the work on roll instabilities to more complicated situations. An obvious starting point is to consider what happens when the pattern that emerges at the primary pattern-forming instability is more complicated – a steady square pattern, for example, or oscillating hexagons, or maybe even a quasipattern. There is an extensive literature dealing with the phase instabilities of steady and oscillatory patterns of all sorts. We shall concentrate on two examples – steady hexagons and steady twelvefold quasipatterns – that illustrate how to extend the methods used in the previous chapter to these harder problems and lead to some interesting new results. At the end of this chapter you will find exercises on the instabilities of steady and oscillating squares as further examples.

Instabilities of hexagons

In this section we will adapt the methods used for rolls to investigate the instabilities of hexagons.