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Effects of boundary conditions on spatially periodic states. (English) Zbl 0599.76099
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.

76Rxx Diffusion and convection
35B32 Bifurcations in context of PDEs
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