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

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.

arXiv comment41 pages, run tex twice

Cited by in corpus (2)

• Modulating traveling fronts for the Swift-Hohenberg equation in the case of an additional conservation law
• Spectral stability of pattern-forming fronts in the complex Ginzburg-Landau equation with a quenching mechanism