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A condensed review is presented for two basic topics in the theory of patternformation in nonlinear dissipative media: (i) domain walls (DWs, alias grainboundaries), which appear as transient layers between different statesoccupying semi-infinite regions, and (ii) two- and three-dimensional (2D and3D) quasiperiodic (QP) patterns, which are built as superpositions ofplane-wave modes with incommensurate spatial periodicities. These topics areselected for the present article, dedicated to the 70th birthday of ProfessorM. 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 maynow seem as "old" ones, they keep their significance as fundamentally importantresults in the theory of nonlinear DW and QP patterns. Adding to the findingsrevealed in the original works by M. I. Tribelsky et al., the present articlealso reports several new analytical results, recently obtained as exactsolutions to systems of coupled real Ginzburg-Landau (GL) equations. These are:a new solution for symmetric DWs in the bimodal system including linear mixingbetween its components; a solution for a strongly asymmetric DWs in the casewhen 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 atrapped bright soliton in the third component; and an exact solution for DWsbetween counter-propagating waves governed by the GL equations withgroup-velocity terms. The significance to the "old" and new results collectedin this review is enhanced by the fact that the systems of coupled equationsfor two- and multicomponent order parameters, addressed in the article, applyequally well to modeling thermal convection, multimode light propagation innonlinear optics, and binary Bose-Einstein condensates.

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