PHASE DYNAMICS OF PATTERNS - THE EFFECT OF BOUNDARY-INDUCED AMPLITUDE VARIATIONS

A generalized phase diffusion equation is derived that incorporates spatial variations of the pattern amplitude. We investigate on the one hand the spatiotemporal relaxation behavior of initially prepared phase perturbations and on the other hand the structure and dynamics of damped phase waves that are forced by time-periodic, spatially localized perturbations. For the two paradigmatic cases of Rayleigh-Benard convection (RBC) in the form of straight parallel rolls and of axisymmetric Taylor vortex flow (TVF), we compare the results of the phase equation for finite setups in quantitative detail with finite-difference numerical simulations of the full two-dimensional hydrodynamic field equations, with Ginzburg-Landau (GL) equations, and with various experiments. The phase equation can be transformed into a Schrodinger-like form with a potential that is determined by the amplitude variations. The free relaxation of phase perturbations is determined by a Sturm-Liouville eigenvalue problem, and the long-time behavior is governed by its lowest positive eigenvalue. This defines an effective diffusion constant D, which is considerably enhanced relative to the reference value D-0 in an ideal system with constant amplitude. Using the GL amplitude profiles one finds that D/D-0 depends only on a specific combination of driving control parameter and system length. Furthermore, one can apply supersymmetry commutation relations to relate the diffusive eigenvalues and eigenmodes of TVF and RBC to each other. For the latter case, the phase equation has a spatially homogeneous phase eigenmode with a zero eigenvalue that admits a free undamped pattern shift as a whole, while inhomogeneities of the phase relax away with higher diffusive eigenmodes. In the full system of equations there appears, instead of the zero-eigenvalue dynamics, a more complicated nondiffusive ultraslow phase dynamics that allows one to reanalyze recent phase diffusion experiments in RBC. Also, the spatially varying decay rates and wave numbers of periodically forced damped phase waves are shown to depend on amplitude variations and the finiteness of the system. We elucidate this dependence and show how these wave characteristics differ from each other and show that they are in general unrelated to the phase diffusion constant.