Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth

We focus on the morphochemical reaction-diffusion model introduced in Bozzini et al. (2013) and carry out a nonlinear bifurcation analysis with the aim to characterize the shape and the amplitude of the patterns arising as the result of Turing instability of the physically relevant equilibrium. We perform a weakly nonlinear multiple scales analysis, and derive the normal form equations governing the amplitude of the patterns. These amplitude equations allow us to construct relevant solutions of the model equations and reveal the presence of multiple branches of stable solutions arising as the result of subcritical bifurcations. Hysteretic type phenomena are highlighted also through numerical simulations. We show the occurrence of spatial pattern propagation and derive the Ginzburg-Landau equation describing the envelope of the traveling wavefront.