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

We focus on the morphochemical reaction-diffusion model introduced in [13] and carry out a nonlinear bifurcation analysis with the aim to characterize the shape and the amplitude of the patterns close to the marginal stability of the physically relevant equilibrium. We perform a weakly nonlinear multiple scales analysis for both the 1D and 2D spatial case, deriving the equations for the amplitude of the stationary patterns. The analysis of such amplitude equations reveals a large number of interesting phenomena for the system as stable supercritical and subcritical Turing patterns and an hysteretic-type phenomenology due to multiple branches of stable solutions arising in the subcritical case. Moreover, in case of large domains, we show that the pattern forms sequentially and that traveling wavefronts are the precursors to patterning.