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

@article{Eckmann2000NonlinearSO,
  title={Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation},
  author={Jean-Pierre Eckmann and Guido Schneider},
  journal={Communications in Mathematical Physics},
  year={2000},
  volume={225},
  pages={361-397}
}
Abstract: We consider front solutions of the Swift–Hohenberg equation ∂tu= -(1+ ∂x2)2u + ɛ2u -u3. 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 ∂tu(x,t) = ∂x2u (x,t)+(1+tanh(x-ct))u(x,t)+u(x,t)p with… 
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