Validity of the Cahn–Hilliard approximation for modulations of slightly unstable pattern in the real Ginzburg–Landau equation

In order to describe slow modulations in time and space of slightly unstable spatially periodic stationary solutions of pattern forming reaction–diffusion systems, the Cahn–Hilliard equation can be derived via multiple scaling analysis as a formal approximation equation. By proving estimates between the approximations obtained via this procedure and the exact solutions of the original system the validity of the Cahn–Hilliard equation as an approximation equation can be rigorously justified. This has been done for a class of one-dimensional reaction–diffusion systems in Düll (2007)  [18]. In this paper, we provide a simpler, more elementary and shorter validity proof for the case of the real Ginzburg–Landau equation as an original pattern forming system by exploiting the S1S1-symmetry of the real Ginzburg–Landau equation.