Search results for " Ginzburg-Landau equation"
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WAVE PROPAGATION AND PATTERN FORMATION FOR A REACTION-DIFFUSION SYSTEM WITH NONLINEAR DIFFUSION
2008
We investigate the formation of macroscopic spatio-temporal structures (patterns) for a reaction-diffusion system with nonlinear diffusion. We show that cross-diffusion effects are responsible of pattern initiation. Through a weakly nonlinear analysis we are able to predict the shape and the amplitude of the pattern. In the weakly nonlinear regime we derive the Ginzburg-Landau equation which captures the envelope evolution and the progressing of the pattern as a wave. Numerical simulations, performed using both a particle and a spectral method, are in good agreement with the analytical results.
Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models
2022
In this work we consider a quite general class of two-species hyperbolic reaction-advection-diffusion system with the main aim of elucidating the role played by inertial effects in the dynamics of oscillatory periodic patterns. To this aim, first, we use linear stability analysis techniques to deduce the conditions under which wave (or oscillatory Turing) instability takes place. Then, we apply multiple-scale weakly nonlinear analysis to determine the equation which rules the spatiotemporal evolution of pattern amplitude close to criticality. This investigation leads to a cubic complex Ginzburg-Landau (CCGL) equation which, owing to the functional dependence of the coefficients here involve…