Phase dynamics in the real Ginzburg-Landau equation

Math. Nachr. 263-264 (2004) 171-180

Ian Melbourne and Guido Schneider

Abstract

Spatially periodic equilibria A(X,T)= (1-q²)^1/2 e^{iqX+ ip₀} are the locally preferred planform for the Ginzburg-Landau equation A_T = A + A_XX - 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_t = [h(q)]_xx.

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.