Mathematical Sciences: Nonlinear Localization

Faris教授的研究课题是波传播的两个前沿领域：随机波和非线性波之间的关系。了解这些类型的波及其相互作用对我们了解基本物理现象以及在各种领域的技术应用都很重要；例如，等离子体中的电磁波，有杂质的光纤，以及掺杂的超晶格。当波所经过的介质是复杂的、不规则的，可以看作是随机介质。描述波在高度随机介质中传播的方程的解被称为随机波，人们刚刚开始充分理解它，以便与实验进行比较。在某些情况下，会出现一种称为局部化的显著现象，即反复散射实际上会捕获波。在局部化和更常见的扩散之间转换的性质还没有被很好地理解，这里的数学进展将是波与物质复杂相互作用研究中的一个概念上的进步。在之前的工作中，Faris教授发现了新的情况，在维度高于1的情况下，定位发生了，尽管这不是唯一可能的行为。在目前的提议中，他在非线性存在的情况下处理局部化。在这种情况下，波与自身相互作用并产生孤波或孤子。这些波在很多方面表现得像经典粒子。目标是获得随机介质中孤子的严格数学结果。这将是理解复杂随机系统动力学的重要一步。

The subject of Professor Faris' research is the relation between two frontier areas in wave propagation: random waves and nonlinear waves. The understanding of these types of waves and their interaction is of importance both for our knowledge of fundamental physical phenomena as well as technological applications in a wide variety of fields; for example, electromagnetic waves in plasma, optical fibers with impurities, and doped superlattices. When the medium through which a wave passes is complicated and irregular, it may be treated as a random medium. The solutions, called random waves, of the equations that describe wave propagation in a highly random medium, are just beginning to be sufficiently well understood to compare with experiment. In some situations a remarkable phenomenon occurs, called localization, in which repeated scattering actually traps a wave. The nature of the transition between localization and the more common case of diffusion is not yet well understood, and mathematical progress here would be a conceptual advance in the study of complicated interactions of waves and matter. In previous work, Professor Faris has found new situations, in dimension higher than one, in which localization takes place despite the fact that this is not the only possible behavior. In the current proposal, he treats localization in the presence of nonlinearities. In this context, the wave interacts with itself and produces solitary waves, or solitons. These waves behave in many ways like classical particles. The goal is to obtain rigorous mathematical results for solitons in random media. This would be a major step in understanding the dynamics of complex random systems.