The amplitude equation for degenerate subcritical bifurcations in pattern formating systems
- G. Schneider
Abstract
We consider a model for a pattern formating system on the infinite line
which leads to a subcritical bifurcation. For the degenerate case
we use multiple scaling analysis to derive
as an amplitude equation for the envelope A of the bifurcating
pattern which is modulated slowly in time and in space.
We show exact estimates between the approximations obtained
via the amplitude equation and true solutions of the original system.
Moreover, we show that every small solution of the original system
develops in such a way that it can be described after a certain time
by the solutions of the amplitude equation.
The difficulty is to show the estimates on an
-time
scale in contrast to
for the classical Ginzburg-Landau equation, if
is the order of the amplitude. This theory allows the description of modulated
N-pulses in the original system.