- 00-116 J.-P. Eckmann, G. Schneider
- Non-linear Stability of Modulated Fronts
for the Swift-Hohenberg Equation
(390K, postscript, 41p)
Mar 16, 00
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Abstract. We consider front solutions of the
$\partial _t u=-(1+\partial _x^2)^2 u +\epsilon ^2 u -u^3$.
These are traveling waves
which leave in their wake a periodic pattern in the laboratory frame.
Using renormalization techniques and a decomposition into Bloch waves,
we show the non-linear stability of these solutions.
It turns out that this problem is closely related to the
question of stability of the trivial solution for the model problem
$\partial _t u(x,t)=\partial _x^2 u(x,t)+(1+\tanh(x-ct))u(x,t)+u(x,t)^p$
with $p>3$. In particular, we show that the instability of the
perturbation ahead of the front is entirely compensated by a diffusive
stabilization which sets in once the perturbation has hit the bulk
behind the front.