PHASE DYNAMICS OF PATTERNS - THE EFFECT OF BOUNDARY-INDUCED AMPLITUDEVARIATIONS

A generalized phase diffusion equation is derived that incorporates sp atial variations of the pattern amplitude. We investigate on the one h and the spatiotemporal relaxation behavior of initially prepared phase perturbations and on the other hand the structure and dynamics of dam ped phase waves that are forced by time-periodic, spatially localized perturbations. For the two paradigmatic cases of Rayleigh-Benard conve ction (RBC) in the form of straight parallel rolls and of axisymmetric Taylor vortex flow (TVF), we compare the results of the phase equatio n for finite setups in quantitative detail with finite-difference nume rical simulations of the full two-dimensional hydrodynamic field equat ions, with Ginzburg-Landau (GL) equations, and with various experiment s. The phase equation can be transformed into a Schrodinger-like form with a potential that is determined by the amplitude variations. The f ree relaxation of phase perturbations is determined by a Sturm-Liouvil le eigenvalue problem, and the long-time behavior is governed by its l owest positive eigenvalue. This defines an effective diffusion constan t D, which is considerably enhanced relative to the reference value D- 0 in an ideal system with constant amplitude. Using the GL amplitude p rofiles one finds that D/D-0 depends only on a specific combination of driving control parameter and system length. Furthermore, one can app ly supersymmetry commutation relations to relate the diffusive eigenva lues 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 e igenmodes. In the full system of equations there appears, instead of t he zero-eigenvalue dynamics, a more complicated nondiffusive ultraslow phase dynamics that allows one to reanalyze recent phase diffusion ex periments in RBC. Also, the spatially varying decay rates and wave num bers of periodically forced damped phase waves are shown to depend on amplitude variations and the finiteness of the system. We elucidate th is dependence and show how these wave characteristics differ from each other and show that they are in general unrelated to the phase diffus ion constant.