Turing patterns and Newell – Whitehead – Segel amplitude equation

Two-dimensional (2D) reaction–diffusion type systems with linear and nonlinear diffusion terms are examined for their behavior when a Turing instability emerges and stationary spatial patterns form. It is shown that a 2D nonlinear analysis for striped patterns leads to the Newell – Whitehead – Segel amplitude equation in which the contribution from spatial derivatives depends only on the linearized diffusion term of the original model. In the absence of this contribution, i.e., for the normal forms, standard methods are used to calculate the coefficients of the equation. (methodological notes)