Alexander Mielke

The Ginzburg-Landau equation in its role as a modulation equation

The Ginzburg--Landau equation (GLe)

d/dt A = (1 + i alpha) Laplace A + R A - (1 + i beta) |A|2 A ,

with A(t,x) in C, t nonnegative, x in Rd, appears in many different contexts, e.g. nonlinear optics with dissipation or the theory of super conductivity. In the case alpha = beta = 0 (GLe) is called the real GLe and otherwise the complex GLe. Moreover, (GLe) serves as a simple mathematical model for studying the transition of regular to turbulent behavior when for delta > 0 fixed the dispersion parameters (alpha, beta) = s (1, -delta) with s >> 1 are considered.

Our main interest lies in the fact that the solutions of the Ginzburg-Landau equation describe the modulations of patterns in a variety of pattern-forming system. This includes the first applications hydrodynamics: the Rayleigh--B&eacut;nard convection and the Taylor--Couette problem. As such that (GLe) can be seen as a normal form or lowest order expansion of a bifurcation equation in the context of a weakly unstable system when continuous spectrum has been moved over the imaginary axis upon changing an external parameter.

We review the classical derivation of (GLe) as a modulation equation of an original partial differential equation d/dt u = L u + N(u) and give the abstract setting which allows to the application of the Ginzburg-Landau formalism which is based on the ansatz

u(t,x) = eps1/2 Real A(eps t, eps1/2 x) ei k · x Phi + O(eps),

where eps > 0 is the distance of the external parameter from its critical value. We then study the mathmatical properties of (GLe) which are relevant for the justification of the formalism. This includes for instance a global semigroup theory in function spaces containing Linfty(Rd). Finally we give theorems which proves that the solutions A of (GLe) inserted into the above ansatz give good approximation for solutions u of the original system.

Alexander Mielke

Mar 05 1998