Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation

Abstract:

We consider front solutions of the Swift–Hohenberg equation ∂ _t u= -(1+ ∂ _x ²)² u + ɛ² 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 ∂ _t u(x,t) = ∂ _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.