Research on Applied Nonlinear Analysis

9704325罗伊特伯德拟议的研究涉及两个应用数学领域，这两个领域都与非线性波传播有关。一个主题涉及一类自由边界(自由界面)问题的研究，这些问题是作为无气燃烧和快速凝固的简化概念模型自然产生的。这些自由边界问题展示了本质上是低维的各种令人惊讶的动力学模式。这些模型问题的定性行为将从解析和数值两方面进行研究。主要的焦点将放在两个空间维度的传播上。另一个主题来自非线性光学领域。它涉及由Maxwell-Bloch偏微分方程组描述的模型激光动力学。所提出的研究的主要目的将是建立失谐和加宽的Maxwell-Bloch方程的适定性，吸引子维度的估计，以及对同宿混沌的严格研究。此外，还计划对激光湍流进行数值研究。拟议的研究是由技术应用推动的。一个直接相关的技术过程是自蔓延高温合成(SHS)。SHS工艺目前正在作为一种制造某些陶瓷和金属合金的新方法而被研究。这一过程的想法是将粉末成分组合在一起，通过向样品发送火焰波来将它们烘焙成所需的产品。与在炉子中焙烧混合物的传统制造方法相比，SHS工艺具有许多优点：更均匀和纯的产品，更短的合成时间，使用更简单和更便宜的设备等。第二个拟议研究课题的来源是激光光学，即气体激光器的基本动力学，具有广泛的应用。拟议中的数学调查的目标有两个。一方面，通过对技术应用中不受欢迎的特殊极端制度的研究，我们建议开发简化的诊断工具，允许预测和避免这些不受欢迎的(混乱)制度。另一方面，对极端制度的研究将进一步加深对支配技术制度的基本机制的理解，从而促进对其更好的控制和利用。这项研究将通过分析方法和计算机模拟相结合的方式进行。分析将有助于勾勒出关键的制度及其逻辑结构，而模拟将揭示不适用于分析方法的细节，而这些细节本身可能导致重要的数学发展。

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.