Dispersive effective models for waves in heterogeneous media
Dispersive effective models for waves in heterogeneous media
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异质介质中波的色散有效模型
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发表时间：
2011
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通讯作者：
A. Lamacz
中科院分区：
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作者：
A. Lamacz
我们从可变系数的一维标量波动方程开始研究波在强异质介质中的长期行为。我们假设系数是周期性的，周期为 e，并且 e > 0 是一个小长度参数。我们的主要结果涉及均质化，包括严格推导两种不同的色散模型。第一个是具有常数系数（包括 e 的幂）的四阶方程。在第二个模型中，通过考虑三阶线性化 Korteweg-de Vries 方程，完全避免了 e 相关性。我们的结果是，两个简化模型都很好地描述了长期行为。我们分析中的一个重要工具是自适应算子，它根据介质的周期结构修改平滑函数。
We study the long-time behavior of waves in a strongly heterogeneous medium, starting from the one-dimensional scalar wave equation with variable coefficients. We assume that the coefficients are periodic with period e and e > 0 is a small length parameter. Our main result concerns homogenization and consists in the rigorous derivation of two different dispersive models. The first is a fourth-order equation with constant coefficients including powers of e. In the second model, the e-dependence is completely avoided by considering a third-order linearized Korteweg–de Vries equation. Our result is that both simplified models describe the long-time behavior well. An essential tool in our analysis is an adaption operator which modifies smooth functions according to the periodic structure of the medium.