基于Bloch分解的周期GKdV波的调制方程及其稳定性
On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions
摘要：
本文补充了Bronski和Johnson以及Johnson和Zumbrun关于广义Korteweg-de Vries方程空间周期行波解的调制稳定性的最新结果。在以前的工作中，它是由严格的埃文斯函数计算表明，正式的慢调制近似导致的Whitham系统准确地描述了长波长扰动的光谱稳定性。在这里，我们重现这一结果，而不参考埃文斯函数使用直接布洛赫展开方法和光谱微扰分析。这种方法的优点也适用于更一般的多周期设置，其中还没有方便计算的埃文斯函数。特别是，我们补充的图片所描述的调制稳定的Bronski和约翰逊通过分析投影到总特征空间分叉从原点附近的原点和零Floquet参数。我们表明，所得到的线性系是等价的，领先的顺序和共轭，到Whitham系统，因此，该系统的特征多项式同意（领先的顺序）通过Evans函数计算得到的线性色散关系。
英文摘要：
In this paper, we complement recent results of Bronski and Johnson and of Johnson and Zumbrun concerning the modulational stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation. In this previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to long wavelength perturbations. Here, we reproduce this result without reference to the Evans function by using direct Bloch-expansion methods and spectral perturbation analysis. This approach has the advantage of applying also in the more general multi-periodic setting where no conveniently computable Evans function is yet devised. In particular, we complement the picture of modulational stability described by Bronski and Johnson by analyzing the projectors onto the total eigenspace bifurcating from the origin in a neighborhood of the origin and zero Floquet parameter. We show the resulting linear system is equivalent, to leading order and up to conjugation, to the Whitham system and that, consequently, the characteristic polynomial of this system agrees (to leading order) with the linearized dispersion relation derived through Evans function calculation.
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