Mathematical Sciences: Analysis of Patterns and Dynamics of Nonlinear Dissipative Systems

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Abstract Xin The first part of the proposed research concerns the global existence and asymptotics of front solutions to the two-dimensional Boussinesq combustion system on finite and infinite vertical strips with unbounded chemical nonlinearities, and arbitrary Lewis, Prandtl and Rayleigh numbers. Numerical methods will be implemented for calculating front speeds and controling chaotic flame fronts. The second part concerns the stability, instability of vortex solutions to Ginzburg-Landau equations and related asymptotic analysis. The third part concerns the dimensional bounds of the universal attractors, qualitative properties of solutions, approximation using reduced equations (such as Swift-Hohenberg like equations), and numerical simulation of the Maxwell-Bloch two level laser systems. The above proposed works are based on problems arising in dynamics of reacting flows, and lasers. Understanding reacting flows is of tremendous practical importance, and is related to industrial chemical processing, energy consumption, as well as biotechnological remediation of environment. In designing internal combustion engines or conducting industrial chemical processes, one of the immediate concerns is to increase the efficiency of fuel burning, and so minimize the waste gas output and reduce air pollution. It is known that well-mixed and strongly stirred reacting fluids have more effective contact area and their reaction efficiency is much higher. On the other hand, stirred fluids behave in an irregular and chaotic manner, and are not easy to control. This motivates us to study the properties of the irregular reacting flows in order to utilize them for our needs. Mathematical models proposed here serve as a starting point for pursuing such an investigation where both analytical and numerical methods can be applied to advance our understanding. Moreover, the techniques we develop can be very useful to other problems, such as bioremediation of polluted aquifer. T here similar irregular or chaotic phenomena exist due to the interaction of chemical species with bacteria growth. Chaotic phenomena also appear in optical systems where people try to produce high power laser output for massively parallel optical computing and processing, image storage, and high power energy sources. It is essential to be able to achieve coherence and control of the laser output. Mathematically, this amounts to studying stability and instability of the special solutions of the laser system, analyzing the structures and finding good approximations of attractors. The two level laser model we propose here is the first step towards this goal. The progress made and the methodology developed in the systematic study of the two level model will greatly help us approach more complicated yet more appealing semiconductor laser systems in the future.