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

We consider front solutions of the Swift-Hohenberg equation ∂ t u = −(1 + ∂ x ) 2 u + ε 2 u − u . 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) = ∂ 2 x u(x, t) + (1 + tanh(x− ct))u(x, t) + u(x, t) 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.