Phase Dynamics of Weakly Unstable Periodic Structures

Nonlinear phase dynamics of weakly unstable two-dimensional periodic patterns is studied. Four distinct physical situations are specifically considered. They correspond to the Eckhaus instability and zig-zag instability occurring in each of propagating and non-propagating patterns. Consequently, four prototype partial differential equations for phase function are obtained. Their derivation is totally based on symmetry- and scaling arguments. A simple interpretation of the origin of nonlinearity is given. Although the main part of the present theory is phenomenological, a more rigorous asymptonic theory is also developed for reaction-diffusion eqations.