Dirk Blömker (Augsburg)
Modulation- or Amplitude-Equations are a universal tool to approximate solutions of
complicated systems like partial or stochastic partial dierential equations (SPDEs) near a change
of stability, when there is no center manifold theory available.
One can rely on the natural separation of time-scales at the bifurcation to show that the solution of
the original equation is well described by the bifurcating pattern with an amplitude that is slowly
modulated in time and also in space, if the underlying domain is suciently large. This amplitude
satis es an equation on the slow time- and space-scale, which is called Amplitude- or Modulation-
Equation.
This is useful to explain qualitatively noise induced pattern formation below the change of stability
and stabilization (i.e. destruction of pattern) due to degenerate additive noise.
The approach is on a formal level well known in the physics literature, and for partial differential
equations on unbounded domains rigorously studied in the last two decades. Although the results
for stochastic equations are far more general, for simplicity of presentation we focus mostly on the
less technical stochastic Swift-Hohenberg equation and as an example the convective instability in
Rayleigh-Benard convection.
