A domain wall between single-mode and bimodal states and its transition to dynamical behavior in inhomogeneous systems

We consider domain walls (DWs) between single-mode and bimodal states that occur in coupled nonlinear diffusion (NLD), real Ginzburg-Landau (RGL), and complex Ginzburg-Landau (CGL) equations with a spatially dependent coupling coefficient. Group-velocity terms are added to the NLD and RGL equations, which breaks the variational structure of these models. In the simplest case of two coupled NLD equations, we reduce the description of stationary configurations to a single second-order ordinary differential equation. We demonstrate analytically that a necessary condition for existence of a stationary DW is that the group-velocity must be below a certain threshold value. Above this threshold, dynamical behavior sets in, which we consider in detail. In the CGL equations, the DW may generate spatio-temporal chaos, depending on the nonlinear dispersion. A spatially dependent coupling coefficient as considered in this paper can be realized at least in two different convection systems: a rotating narrow annulus supporting two traveling-wave wall modes, and a large-aspect-ratio system with poor heat conductivity at the lateral boundaries, where the two phases separated by the DW are rolls and square cells.