Phase dynamics with drift: boundary effects

The description of patterns in nonequilibrium systems is a fascinating subject. It becomes somewhat untractable if one insists to keep the original equations in their complete form, as the Oberbeck–Boussinesq equations for instance in Rayleigh–Bénard convection. Various reduction scheme have been imagined, the ultimate one being the phase picture. I examine a simple version of the phase equation, relevant for systems of travelling rolls or for distributed self-oscillations. The boundary effects limit the ability of this phase to drift freely and yields a well-defined space structure, which can be analyzed in two different limits.