Effects of boundary conditions on spatially periodic states

We investigate analytically and numerically the influence of linear homogeneous boundary conditions on the stationary solutions of a simple model for cellular pattern formation in one dimension. For all boundary conditions there exists in a reduced wavenumber band at least one static solution where the amplitude falls below its bulk value near the boundary (“Type-I” solution). A linear stability analysis of the uniform state at threshold reveals that Type-I solutions are often unstable. Then there exists in the full Eckhaus-stable band, a static solution where the amplitude rises above its bulk value near the boundary (“Type-II” solution), or a limit-cycle solution where the amplitude near the boundary oscillates. These solutions bifurcate from the homogeneous state below the bulk threshold and therefore remain finite at threshold.