J. Rougemont
Dynamics of kinks in the Ginzburg-Landau equation:
Approach to a metastable shape and collapse of embedded pairs of kinks
(723K, postscript)
ABSTRACT. We consider initial data for the real Ginzburg-Landau
equation having two widely separated zeros. We
require these initial conditions to be locally close to a stationary
solution (the ``kink'' solution) except for a perturbation supported
in a small interval between the two kinks.
We show that such a perturbation vanishes on a time scale
much shorter than the time scale for the motion of the
kinks. The consequences of this bound, in the context of earlier
studies of the dynamics of kinks in the
Ginzburg-Landau equation, [ER], are as follows: we consider
initial conditions $v_0$ whose restriction to a bounded interval $I$
have several zeros, not too regularly spaced, and other zeros of $v_0$
are very far from $I$. We show that all these zeros eventually
disappear by colliding with each other. This relaxation process
is very slow: it takes a time of order exponential of the length of
$I$.