Domain dynamics in the anisotropic Swift-Hohenberg equation

Two types of asymptotic ordering processes in the anisotropic Swift-Hohenberg equation are studied, paying particular attention to the interaction between domain walls. For the first type, we will discuss the time evolution in which the spatially oscillatory patterns are formed, and show that two kinds of patterns exist depending on whether or not the imaginary part of the field vanishes. When the imaginary part is present, the equation has two distinct states which are regarded as kinds of domains, so the dynamics between two domain walls is established. We then discuss, for the second type, the dynamics when nontrivial uniform states are constructed. There exist two different domain walls, the Ne el type wall and the Bloch type wall, in a similar way to the anisotropic Ginzburg-Landau equation. The equation of motion for two domain walls is derived, and it is shown that the distance between the two domain walls eventually approaches a finite length. The theoretical result is confirmed by numerical simulations. This fact proves the validity of the prediction on the temporal development of the distance between two domain walls.