  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 $L^1\cap L^\infty$, with sharp rates.
