Mathematical Sciences: Nonlinear Localization

法里斯教授的研究主题是 在波浪传播的两个前沿领域之间：随机波和 非线性波 对这些类型的波的理解， 他们的互动对于我们了解 基本的物理现象和技术 广泛应用于各种领域;例如， 等离子体中的电磁波，带有杂质的光纤， 和掺杂的超晶格。 当波通过的介质很复杂时 和不规则，它可以被视为一个随机介质。的 方程的解，称为随机波， 波在高度随机介质中的传播， 充分理解，以与实验进行比较。 在 在某些情况下，会发生一种显著的现象，称为 局部化，其中重复的散射实际上捕获了波。 本地化和更多的过渡性质 扩散的常见情况尚未得到很好的理解， 数学上的进步将是一个概念上的进步， 研究波与物质的复杂相互作用。 在 在以前的工作中，法里斯教授发现了新的情况， 高于1的维度，其中发生局部化 尽管这不是唯一可能的行为。 在 目前的建议，他认为本地化的存在， 非线性 在这种背景下，波与自身相互作用 并产生孤波或孤子。 这些波的行为 在很多方面类似于经典粒子。 我们的目标是获得 随机介质中孤子的严格数学结果。 这 将是理解复杂的 随机系统

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.