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.
Gambino, G., Lombardo, M.C., Sammartino, M. (2008). WAVE PROPAGATION AND PATTERN FORMATION FOR A REACTION-DIFFUSION SYSTEM WITH NONLINEAR DIFFUSION.