三维振动介质中的非均匀性
Inhomogeneities in 3 dimensional oscillatory media
摘要：
以复Ginzburg-Landau方程为原型，研究三维空间均匀振荡的局域扰动。特别是，我们将专注于局部改变振荡相位的不均匀性。在通常的平移不变空间和$ \varepsilon=0$上，对这些空间齐次解进行线性化，得到一个本征值为零的算子嵌入本质谱中。相反，我们表明，当被认为是一个Kondratiev空间之间的操作，线性化是一个弗雷德霍姆算子。这些空间由代数局部化的函数组成，这些函数随每个导数而增加。我们利用这个结果通过隐函数定理构造接近平衡的解，并推导出远场波数的渐近性。
英文摘要：
We consider localized perturbations to spatially homogeneous oscillations in dimension 3 using the complex Ginzburg-Landau equation as a prototype. In particular, we will focus on heterogeneities that locally change the phase of the oscillations. In the usual translation invariant spaces and at $ \varepsilon=0$ the linearization about these spatially homogeneous solutions result in an operator with zero eigenvalue embedded in the essential spectrum. In contrast, we show that when considered as an operator between Kondratiev spaces, the linearization is a Fredholm operator. These spaces consist of functions with algebraical localization that increases with each derivative. We use this result to construct solutions close to the equilibrium via the Implicit Function Theorem and derive asymptotics for wavenumbers in the far field.
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