Turing Instability and Pattern Formation for the Lengyel-Epstein System with Nonlinear Diffusion

In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel-Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case. Moreover, we compute the complex Ginzburg-Landau equation in the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution.