ABSTRACT
The growth of a spatially structured mode at an instability point of a uniform state is a fundamental mechanism of pattern formation. As discussed in Section 5.3.3, there are two general cases leading to the growth of spatially structured modes, a non-uniform stationary instability leading to stationary periodic patterns, and a non-uniform oscillatory instability leading to traveling-wave patterns. We use here the amplitude equation approach presented in the previous chapter to analyze these two types of patterns close to their onset. In Section 7.1.1 we consider a variant of the SH equation that breaks the inversion symmetry u → −u and allows for stationary hexagonal patterns in addition to stripe patterns. Using symmetry consideration we motivate the form of the amplitude equations and use them to study two types of hexagonal patterns and the interaction between hexagonal and stripe patterns.
In Section 7.1.2 we consider the amplitude equations for a pair of traveling waves propagating in opposite directions, and study the wave patterns that can result from the interaction between the two.
