Modulation Equations for Spatially Periodic Systems: Derivation and Solutions

Abstract

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 the parameter space, we derive a coupled system of amplitude equations. This can either be reduced to an equation for a real amplitude with cubic and quadratic terms or it can be written as a Ginzburg--Landau equation for a complex amplitude A, with an additional term, proportional to $\ol{A}$. For this latter equation, we study the existence and stability of periodic solutions. We find that the nonsymmetric term $\ol{A}$ decreases the width of the Eckhaus band of stable solutions. Numerical simulations show that complex periodic solutions bifurcate into stable, real solutions for increasing influence of the $\ol{A}$-term.