Mathematical Sciences: Waves and Diffusion in Random Media

该研究包括三个部分：(A)随机层状介质中声脉冲反射的正问题和反问题的研究；(B)非线性薛定谔方程和相关方程聚焦奇异性的研究；(C)对流增强扩散问题的研究。在脉冲反射中，将为由位于表面或随机分层的半空间上方的点源发射的脉冲产生的反射信号开发统计理论。该理论基于随机方程的渐近理论，将通过数值模拟来探索其有效性范围。对于入射到随机分层半空间上的脉冲平面波，将建立和求解一系列统计反问题，其中介质的性质基于观测到的反射信号的统计。脉冲反射方面的工作直接关系到勘探地震学，在较小程度上与探测自然或人为地震事件有关。例如，关于非线性薛定谔方程的工作在激光外科的非线性光学和激光聚变中是有用的。对流扩散方面的工作有助于研究污染物在海洋和大气等各种环境中的扩散。

The proposed research has three components: (a) The study of direct and inverse problems for acoustic pulse reflection from randomly layered media, (b) the study of the focusing singularity of the nonlinear Schroedinger and related equations, and (c) convection enhanced diffusion. In pulse reflection a statistical theory will be developed for reflected signals generated by pulses emitted by a point source located on the surface or above a randomly layered half space. The range of validity of the theory, based on the asymptotic theory of stochastic equations, will be explored with numerical simulations. For pulsed plane waves incident on a randomly layered half space, a number of statistical inverse problems will be formulated and solved, where properties of the medium are based on the statistics of the observed reflected signals. The work on pulse reflection is of direct interest in exploration seismology and to a lesser extent in connection with the detection of seismic events that are natural or man-made. The work on the nonlinear Schroedinger equation is useful in nonlinear optics for laser surgery, for example, and in laser fusion. The work on convection-diffusion is useful in the study of dispersion of pollutants in various environments such as the ocean and the atmosphere.