Past and present trends in the development of the pattern-formation theory: domain walls and quasicrystals

Abstract

A condensed review is presented for two basic topics in the theory of pattern formation in nonlinear dissipative media: (i) domain walls (DWs, alias grain boundaries), which appear as transient layers between different states occupying semi-infinite regions, and (ii) two-and three-dimensional (2D and 3D) quasiperiodic (QP) patterns, which are built as superposition of plane-wave modes with incommensurate spatial periodicities. These topics are selected for the present article, dedicated to the 70th birthday of Professor M. I. Tribelsky, due to the impact made on them by papers published by Prof. Tribelsky and his coauthors. Although some findings revealed in those works may now seem as "old" ones, they keep their significance as fundamentally important results in the the theory of nonlinear DW and QP patterns. Adding to the findings revealed in the original works by M. I. Tribelsky et al., the present article also reports several new analytical results, obtained as exact solutions to systems of coupled real Ginzburg-Landau (GL) equations. These are: a new solution for symmetric DWs in the bimodal system including linear mixing between its components; a solution for a strongly asymmetric DWs in the case when the diffusion (second-derivative) term is present only in one GL equation; a solution for a system of three real GL equations, for the symmetric DW with a trapped bright soliton in the third component; and an exact solution for DWs between counter-propagating waves governed by the GL equations with group-velocity terms. The significance to the "old" and new results collected in this review is enhanced by the fact that the systems of coupled equations for two-and multicomponent order parameters, addressed in the article, apply equally well to modeling thermal convection, multimode light propagation in nonlinear optics, and binary Bose-Einstein condensates.