Grain boundaries in two-dimensional traveling-wave patterns

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Results of systematic numerical studies of grain boundaries (GBs) between traveling waves are reported. Starting from the two-dimensional (2D) complex Swift-Hohenberg (SH) equation, we demonstrate that it admits steadily moving GBs in the form of sinks and sources. The sinks are always stable, while the sources may be stable or unstable. Sometimes, chaotic patterns are also observed. Next, we reduce the 2D SH equation to two coupled 1D SH equations, assuming that the wave field is a superposition of two waves with fixed y-components of the corresponding wave vectors. Direct comparison demonstrates that typical wave patterns described by the 2D equation and by the coupled 1D equations are practically indistinguishable. Within the framework of the latter approximation, we compute dependence of velocities of the sinks and sources upon the orientation of the traveling waves. In particular, for nearly equal wave vectors on both sides of GB, the velocity proves to be close to the mean group velocity of the pattern. In the opposite limit of nearly antiparallel wave vectors, the sink's velocity is close to an earlier analytical prediction (two thirds of the mean group velocity). We have also considered further simplifications of the coupled 1D SH equations, reducing them to coupled Ginzburg-Landau equations, and then to a nonlinear phase-diffusion equation. The simplified equations still provide a satisfactory description of various dynamical regimes. Especially, the stability of the stationary source is discussed with the nonlinear phase-diffusion equation. Finally, we report preliminary results of simulations of patterns containing triple points, produced by collisions between nonparallel GBs. The triple point may be both stable and unstable.