Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation
Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation
复制标题
Swift-Hohenberg 方程的调制前沿的非线性稳定性¶
DOI：
10.1007/s002200100577
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发表时间：
2000
影响因子：
2.4
通讯作者：
G. Schneider
中科院分区：
文献类型：
--
作者：
J. Eckmann;G. Schneider
摘要：我们考虑Swift-Hohenberg方程∂tu= -(1+∂x2)2u + æ 2u -u3的前解。这些行波在它们的尾迹中留下一个实验室框架中的周期性图案。利用重整化技术和分解成布洛赫波，我们证明了这些解的非线性稳定性。事实证明，这个问题与模型问题∂tu(x,t) =∂x2u (x,t)+(1+tanh(x-ct))u(x,t)+u(x,t)p with p>3的平凡解的稳定性问题密切相关。特别地，我们证明了锋面前方扰动的不稳定性完全由扩散补偿
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 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.