### 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 R**^{d},
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) = eps**^{1/2} Real A(eps t, eps^{1/2} x)
e^{i 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
**L**^{infty}(R^{d}). 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