Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media
Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media
复制标题
随机介质中波的分数白噪声极限和近轴逼近
DOI：
--
复制
发表时间：
2017
影响因子：
2.5
通讯作者：
O. Pinaud
中科院分区：
文献类型：
--
作者：
Christophe Gomez;O. Pinaud
这项工作致力于具有长程相关性的随机介质中高频波传播的渐近分析。我们对两个渐近状态感兴趣，我们同时研究它们：近轴近似，其中波被准直并沿着特定的传播方向传播，以及白噪声极限，其中背景中的随机波动在统计意义上可以通过分数白噪声很好地近似。波动的分数性质让人想起基础随机介质中的长期相关性。典型的物理环境是激光束在湍流大气中的传播。从速度场中快速非高斯随机振荡的高频波动方程出发，我们推导了分数阶伊藤-薛定谔方程，即势能等于分数白噪声的薛定谔方程。证明涉及对反向散射以及传播模式和倏逝模式之间的耦合的精细分析。由于长程依赖性，具有随机系数的方程的经典扩散近似定理不适用，因此我们使用矩技术来研究收敛性。
This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Itô–Schrödinger equation, that is, a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.