Reaction and Concentration Dependent Diffusion Model

We study the formation of patterns in the genuinely nonlinear reaction diffusion model equation $u_t+2a(u^2{)}_x=(u^2{)}_{\mathrm{xx}}+F(x,u)$, where $u$ may be viewed as an amplitude of a thermal wave in plasma or density of a biological species and $F=u(1-u)$ or $F=q\left(x\right)u^l$, $l=0,2\mathrm{}$. We provide a transformation which maps the model into a purely diffusive process free of its interacting part and its intrinsic temporal and spatial scales. The well known attractors of the diffusive process enable us to completely characterize the emerging patterns which, depending on $F$ and initialization, may be a semicompact, or a compact, traveling wave or a nontrivial equilibrium.