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 Swift-Hohenberg equation $\partial _t u=-(1+\partial _x^2)^2 u +\epsilon ^2 u -u^3$. 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.

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