Progress of Theoretical Physics, Vol. 71, No.5, May 1984 Sum Rule and Scaling Law of Nonlinear Relaxation of Repulsively Interacting Kinks Ken SEKIMOTO . Research Institute for Funda,mental Physics Kyoto University, Kyoto 606 (Received January 20, 1984) 885 We study the nonlinear relaxation process of the repulsively interacting kinks which were initially arrayed almost periodically in a semi-infinite space. The motion of the kinks toward the remaining semi- infinite space is analysed using the time-dependent-Ginzburg-Landau equation. The main results are; (j) the sum of the displacements of the kinks increases asymptotically linearly in time, and (ji) if the inter-kink interaction satisfies certain conditions, the kink-density function obeys asymptotically the scaling law in which the characteristic length increases as t"2. § 1. Introduction Recently we have more and more appreciated the important· roles played by the topological defects in the systems undergoing the first order phase transition_ .-, There are some successful theories on the dynamics of the first order phase transition in which the motion of the topological defects, such as domain walls 1) and kinks,2) is explicitly consider- ed. These theories are based on the more fundamental work in which the equations of motion of those topological defects are derived from that of the underlying order-param- eter fields. 3 ),4) While the detailed aspects of the motion of the topological defects depend on the original field 'theoretical model, the qualitative nature of the phase transition is often insensitive to the details of the motion of the individual topological defects. . In this paper we discuss a certain nonlinear relaxation process of the system which contains the repulsively interacting kinks, and we show that there are some aspects of the process which do not depend on the details of the interaction between the kinks. In §2 we describe the model and summarize the results. The details of their deriva- tion are given in §3. The last section (§4) is for the discussion. § 2. The model and the main results We describe the model which we will study. Suppose that there is the system (along the x -axis) which bears an infinite number of kinks, the topological defec;ts on the 1- dimensional field with discrete symmetry. We denote by Xn(t) (n=O, 1, 2, ... ) the position of the n-th kink at time t, where we assume that xnCt )<Xn+1Ct). Throughout this paper we assume that both the length unit and the time unit are made dimension-less to simplify the notation. The effective interaction energy between the kinks would be derived from the proper microscopic model (see, for example, Ref. 5)). Generally the interaction .. ' between the neighboring kinks (the inter-kink interaction for brevity) is repulsive, which is in contrast with the kink-antikink interaction, and is known to have the following asymptotic form, (2·1) Downloaded from https://academic.oup.com/ptp/article-abstract/71/5/885/1874908 by guest on 16 June 2020