Phase dynamics in the real Ginzburg-Landau equation

Spatially periodic equilibria A(X, T) = 1 - q2 eiqX+i0 are the locally preferred planform for the Ginzburg-Landau equation TA = 2XA + A - A|A|2. To describe the global spatial behavior, an evolution equation for the local wave number q can be derived formally. The local wave number q satisfies approximately a so called phase diffusion equation q = 2h(q). It is the purpose of this paper to explain the extent to which the phase diffusion equation is valid by proving estimates for this formal approximation.