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

本文的第一部分研究了二维Boussinesq燃烧系统在有限条和无限条上的整体解的存在性和渐近性，该系统具有无界的化学非线性和任意的Lewis、Prandtl和Rayleigh数。将采用数值方法计算前锋速度和控制混沌火焰前锋。第二部分讨论了Ginzburg-Landau方程涡旋解的稳定性、不稳定性以及相关的渐近分析。第三部分讨论了万能吸引子的维界、解的定性性质、约化方程(如Swift-Hohenberg方程)的逼近以及Maxwell-Bloch二能级激光系统的数值模拟。上述工作是基于反应流动力学和激光动力学中出现的问题。了解反应流具有非常重要的实际意义，涉及到工业化学加工、能源消耗以及环境的生物技术修复。在设计内燃机或进行工业化学过程时，当务之急之一是提高燃料燃烧效率，从而将废气排放降至最低，减少空气污染。众所周知，混合良好、搅拌强烈的反应液具有更大的有效接触面积，其反应效率也更高。另一方面，搅拌后的流体表现出不规则和混乱的行为，不容易控制。这促使我们研究不规则反应流的性质，以便利用它们来满足我们的需要。这里提出的数学模型是进行这种研究的起点，其中分析和数值方法都可以用来促进我们的理解。此外，我们开发的技术对其他问题也非常有用，例如受污染的含水层的生物修复。由于化学物种与细菌生长的相互作用，也存在类似的不规则或混沌现象。混沌现象也出现在光学系统中，人们试图产生高功率激光输出，用于大规模并行光学计算和处理、图像存储和高功率能源。能够实现激光输出的相干和控制是至关重要的。从数学上讲，这相当于研究激光系统特解的稳定性和不稳定性，分析系统的结构，找到吸引子的良好逼近。我们在这里提出的二能级激光模型是迈向这一目标的第一步。在系统研究二能级模型方面取得的进展和发展的方法论将极大地帮助我们在未来接近更复杂但更有吸引力的半导体激光系统。

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.