Solutions for nonlinear convection in the presence of a lateral boundary

A method is described for calculating steady-state patterns in large-scale nonlinear systems, taking into account the effect of a lateral boundary and without the need for extensive numerical calculations. The key feature is the determination of the phase shift of the nonlinear periodic form distant from the boundary as a function of wavelength. This is found by analyzing the solution close to the boundary, where Floquet theory is used to describe the departure of the solution from its periodic form. For a restricted band of wavelengths lying within the Eckhaus boundary, dual solutions for the phase shift are found, one of which corresponds to an unstable state. Results are presented here for the one-dimensional Swift-Hohenberg equation in a semi-infinite domain but in principle the method can be applied to more general pattern-forming systems. The results are compared with the predictions of weakly nonlinear theory and with nonlinear computations on a large but finite domain.