Dynamics of kinks in the Ginzburg-Landau equation: Approach to a metastable shape and collapse of embedded pairs of kinks

This paper is a continuation of [ER], where a model of interface dynamics was analyzed. This model is based on the Ginzburg-Landau equation in an unbounded one-dimensional domain. A similar model had originally been studied on a finite interval subject to Neumann boundary conditions by J. Carr and R.L. Pego, [CP1,CP2]. For a physical motivation and a discussion of related models, see Bray, [B], and references therein. The interfaces are defined as the zeros of a solution v(x, t) of the real Ginzburg-Landau evolution equation. These zeros are shown to have the following behavior: let their positions on the real line be denoted by zk(t), with zj(t) < zj+1(t), j = 0, . . . , N − 1. When the zeros are sufficiently far from each other, their dynamics is approximately described by: