Research on Applied Nonlinear Analysis

小行星9704325 拟议的调查解决两个领域的应用数学， 都与非线性波传播有关。其中一个主题涉及 研究一类自由边界（自由界面）问题， 自然作为简化的概念模型的无气燃烧和快速 固化这些自由边界问题展示了一个惊人的 各种基本上是低维的动力学模式。的 这些模型问题的定性行为将被调查， 分析和数字。主要重点将放在传播， 二维空间另一个话题是从非线性领域引出的 光学.它涉及的模型激光动力学是由 Maxwell-Bloch偏微分方程组的主旨 拟议的研究将是建立适定性的 具有失谐和加宽的Maxwell-Bloch方程， 吸引子维数和同宿混沌的严格研究。一 激光湍流的数值研究也在计划之中。 这项研究的动机是技术应用。一 直接相关的工艺过程是自蔓延高压 高温合成（SHS）。SHS进程目前正在 研究作为一种新的方法，制造某些陶瓷和金属 合金.该方法的想法是将粉末状成分联合收割机和 通过将火焰波发送到烤成所需的产品， sample.自蔓延高温合成工艺与传统工艺相比具有许多优点， 在熔炉中烘烤混合物的制造方法：更多 均匀和纯净的产品，更短的合成时间，使用更简单和 廉价的设备等第二个提出研究课题的来源 在于激光光学，即气体激光器的基本动力学， 各种应用程序所提出的数学目标 调查是双重的。 一方面，通过对特殊极端的研究， 在技术应用中不受欢迎的制度，我们建议 开发简化的诊断工具，以预测和避免这些 不受欢迎的（混乱的）制度。另一方面，研究极端 制度将促进更好地了解基本机制， 管理技术制度，因此将促进他们更好地 控制和利用。这项研究将通过一个 分析方法和计算机模拟相结合。分析 将有助于界定重要的制度及其逻辑结构， 模拟将揭示不适合分析的细节， 方法，并在自己的回合，可以导致重要的数学 发展理念

9704325 Roytburd The proposed investigations address two areas of applied mathematics that are both related to nonlinear wave propagation. One topic concerns the study of a class of free boundary (free interface) problems which arise naturally as simplified conceptual models of gasless combustion and fast solidification. These free boundary problems demonstrate an amazing variety of dynamical patterns that are essentially low-dimensional. The qualitative behavior of these model problems will be investigated both analytically and numerically. The principal focus will be on propagation in two spatial dimensions. Another topic is drawn from the field of nonlinear optics. It concerns the model laser dynamics which is described by the Maxwell-Bloch system of partial differential equations. The main thrust of the proposed research will be at establishing well-posedness for the Maxwell-Bloch equations with detuning and broadening, evaluation of the attractor dimension, and the rigorous study of the homoclinic chaos. A numerical investigation of laser turbulence is also planned. The proposed research is motivated by technological applications. One immediately related technological process is the self-propagating high temperature synthesis (SHS). The SHS process is currently being investigated as a new method of manufacturing certain ceramic and metallic alloys. The idea of the process is to combine the powdered ingredients and bake them into the desired product by sending a flame wave through the sample. The SHS process has a number of advantages over the traditional manufacturing methods in which the mixture is baked in a furnace: more uniform and pure products, shorter synthesis times, use of simpler and cheaper equipment etc. The source of the second proposed research topic lies in laser optics, namely basic dynamics of gaseous lasers that find a wide variety of applications. The goal of the proposed mathematical investigations is twofold. On one hand, through the study of special extreme regimes which are undesirable in technological applications, we propose to develop simplified diagnostic tools that will allow to predict and avoid these undesirable (chaotic) regimes. On the other hand, the study of the extreme regimes will further the better understanding of basic mechanisms that govern the technological regimes and therefore will promote their better control and utilization. The research will be conducted through a combination of analytical methods and computer simulations. The analysis will help to delineate crucial regimes and their logical structure, while the simulations will uncover details which are not amenable to analytical methods and, in their own turn, can lead to important mathematical developments.