Stable stationary solutions in reaction-diffusion systems consisting of a 1-D array of bistable cells

In this paper we present the construction of stable stationary solutions in reaction-diffusion systems consisting of a 1-D array of bistable cells with a cubic nonlinearity and with a cubic-like piecewise-linear nonlinearity. Some periodic solutions, kinks, solitons are considered. While it is known that spatial chaos arises in such systems with small coupling constants, we show the existence of spatial chaos for an arbitrary value of the cell coupling constant, in the case of the piecewise-linear nonlinearity. The value of the spatial entropy is found. We also show the existence of stable spatially periodic (pattern) solutions that persist for large coupling constants.