The linear stability analysis of Chapter 2 predicts that a small perturbation about a uniform state will grow exponentially in magnitude when the uniform state becomes unstable. Over time, the magnitude of a perturbation will grow so large that the nonlinear terms that were neglected when deriving the linearized evolution equation can no longer be ignored. These nonlinear terms play a fundamental role in the resulting pattern formation: they saturate the exponential growth, and they select among different spatial states. It is the essential role of nonlinearity in a spatially extended system that makes the study of pattern formation novel and hard.

We can gain a great deal of insight about the nonlinear regime of pattern formation by considering spatially periodic patterns. This is natural when considering the fate of a single exponentially growing Fourier mode of the linearized evolution equations associated with a linear stability analysis. Nonlinearities in the evolution equations for the system generate spatial harmonics (Fourier modes with wave vectors nq with n an integer) of this growing mode so that the finite-amplitude solution maintains the periodicity over the length 2π/q. A key role of the nonlinearity is to quench the exponential growth of the solution, leading to steady spatially periodic solutions for a stationary instability, and nonlinear oscillations or waves for an oscillatory instability. If this steady or periodic solution is to be physically relevant, we must also require that it be stable with respect to small perturbations. Thus we will study the existence and stability of steady or oscillatory spatially periodic (for qc ≠ 0) solutions.