Diffusive Stability of Turing Patterns via Normal Forms

We investigate dynamics near Turing patterns in reaction–diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a “normal form” coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction–diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to perturbations that are small in (Formula presented.), with sharp rates, recovering and slightly improving on results in Johnson and Zumbrun (Ann Inst H Poincaré Anal Non Linéaire 28:471–483, 2011) and Schneider (Commun Math Phys 178:679–702, 1996).