Pattern Formation - Analysis

Analysis

The analysis of pattern-forming systems often consists of finding a Partial differential equation model of the system (the Swift-Hohenberg equation is one such model) of the form

where F is generically a nonlinear differential operator, and postulating solutions of the form

where the are complex amplitudes associated to different modes in the solution and the are the wave-vectors associated to a lattice, e.g. a square or hexagonal lattice in two dimensions. There is in general no rigorous justification for this restriction to a lattice.

Symmetry considerations can now be taken into account, and evolution equations obtained for the complex amplitudes governing the solution. This reduction puts the problem into the form of a system of first-order Ordinary differential equation, which can be analysed using standard methods (see dynamical systems). The same formalism can also be used to analyse bifurcations in pattern-forming systems, for example to analyse the formation of convection rolls in a Rayleigh-Bénard experiment as the temperature is increased.

Such analysis predicts many of the quantitative features of such experiments - for example, the ODE reduction predicts hysteresis in convection experiments as patterns of rolls and hexagons compete for stability. The same hysteresis has been observed experimentally.