Past and Present Trends in the Development of the Pattern-Formation Theory: Domain Walls and Quasicrystals

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 a superposition of plane–wave modes with incommensurate spatial periodicities. These topics are selected for the present review, dedicated to the 70th birthday of Professor Michael I. Tribelsky, due to the impact made on them by papers of Prof. Tribelsky and his coauthors. Although some findings revealed in those works may now seem “old”, they keep their significance as fundamentally important results in the theory of nonlinear DW and QP patterns. Adding to the findings revealed in the original papers by M.I. Tribelsky et al., the present review 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 of 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 this review, apply equally well to modeling thermal convection, multimode light propagation in nonlinear optics, and binary Bose–Einstein condensates.