WAVE-NUMBER SELECTION IN CONVECTION AND RELATED PROBLEMS

The wave number selection process induced by grain boundaries is studied beyond the framework of amplitude equations. It is shown that the process is an intrinsic property of the problem, being associated with its spatial symmetry. The process always results in discretization of possible values of the wave number. If the control parameter is small enough (so that pinning of the grain boundary to the small scale underlying structure may be neglected) the system possesses an extra restricting condition: the Lyapunov functional density for both coexisting structures must coincide. In this case the selected wave number provides a local minimum of the Lyapunov functional for the spatially uniform patterns. As an application of the theory we consider an extended Swift-Hohenberg equation for rolls with subcritical bifurcation.