Non-linear wavelength selection for pattern-forming systems in channels

A method is described for calculating non-linear steady-state patterns in channels taking into account the effect of an end wall across the channel. The key feature is the determination of the phase shift of the non-linear periodic form distant from the end wall as a function of wavelength. This is found by analysing the solution close to the end wall, where Floquet theory is used to describe the departure of the solution from its periodic form and to locate the Eckhaus stability boundary. A restricted band of wavelengths is identified, within which solutions for the phase shift are found by numerical computation in the fully non-linear regime and by asymptotic analysis in the weakly non-linear regime. Results are presented here for the 2D Swift–Hohenberg equation but, in principle, the method can be applied to more general pattern-forming systems. Near onset, it is shown that for channel widths less than a certain critical value, the restricted band includes wavelengths shorter and longer than the critical wavelength, whereas for wider channels only shorter wavelengths are allowed.