Research on Applied Nonlinear Analysis

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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.