Modulation Equations for Spatially Periodic Systems: Derivation and Solutions

We study a class of partial differential equations in one spatial dimension, which can be seen as model equations for the analysis of pattern formation in physical systems defined on unbounded, weakly oscillating domains. We perform a linear and weakly nonlinear stability analysis for solutions that bifurcate from a basic state. The analysis depends strongly on the wavenumber p of the periodic boundary. For specific values of p, which are called resonant, some unexpected phenomena are encountered. The neutral stability curve which can be derived for the unperturbed, straight problem splits in the neighborhood of the minimum into two, which indicates that there are two amplitudes involved in the bifurcating solutions, each one related to one of the minima. The character of the modulation equation, which describes the nonlinear evolution of perturbations of the basic state, depends crucially on the distance of the bifurcation parameter from the lowest, most critical minimum. In a relatively large part of th...