Dynamical systems theory and singular perturbation analysis for patterns, bubbles, and chemical reduction methods

本项目涉及具有多个长度和时间尺度的系统，目的是分析最近的实验，改进计算方法，并为这些系统建立新的数学理论。首先，在多长度尺度的化学模式领域，最近发现的自复制点和脉冲现象对偏微分方程解的建模和稳定性分析提出了新的挑战。研究者和合作者建立在他们的动力学，时间尺度和机制的分析，负责自我复制，以研究组织自我复制制度的潜在分岔层次结构，进一步研究非局部特征值问题稳定性分析中的零极抵消现象，推广重整化群技术以建立脉冲的全非线性稳定性，并将其推广到两个以上长度尺度的系统。其次，在日益重要的多时间尺度大化学反应系统的还原方法领域，分析了某些方法的有效性和准确性，特别关注Lam和Goussis的计算奇异摄动方法。第三，分析了Oya-Vallochi地下生物修复模型。生物修复是指微生物在电子受体的存在下，降解对环境有害的有机化合物的过程。研究者研究了生物活性的行波和平流与弥散。第四，对牛顿流体中气泡的非球形变形进行了基础研究。最后，讨论了非线性薛定谔方程在二维和四维空间上自相似爆破解的存在性问题。本项目涉及的是当前在生物学、化学、工程学和物理学中重要的数学问题，这些问题表现出快速和缓慢的动态过程。首先，与合作者和一名博士生一起，研究者分析了用于模拟生物化学、燃烧和空气污染工程中大型复杂反应系统的计算方法。这些过程，如某些蛋白质的产生、天然气的燃烧和大气中氧化亚氮的形成，通常涉及几百种物质，每一种物质都参与几次反应，反应时间从纳秒到毫秒，甚至几分钟不等。降低系统复杂性的方法，同时保持所需的准确性，对于这些过程的建模是至关重要的。研究者的目的是表明有一种高度精确的方法可以用来提高其他广泛使用的方法的准确性，这些方法嵌入在主要的计算机代码中。其次，研究者和博士生研究生物修复的数学模型，其中微生物被用来降解对环境有害的有机化合物。数学提供了一种有利的方法来确定重要的量，如生物活动区通过湿土柱传播的波速，以及这种速度如何依赖于许多物理参数。第三，对水中气泡动力学进行了基础研究。球形气泡的变形导致振荡的时间尺度比球形模式本身振荡的时间尺度短得多，主要目标是模拟球形和非球形模式之间的非线性能量传递，这种传递可以导致气泡空化和随之而来的水声产生，例如涡轮叶片。最后，研究人员进一步发展了自我复制的化学模式和控制非线性波传播的非原型方程的理论。

Kaper0306523 This project concerns systems with multiple length and timescales, with the goals of analyzing recent experiments, ofimproving computational methods, and of establishing newmathematical theory for such systems. First, in the area ofchemical patterns with multiple length scales, therecently-discovered phenomena of self-replicating spots andpulses has posed new challenges for modeling and for stabilityanalysis of solutions of partial differential equations. Theinvestigator and collaborators build on their analysis of thedynamics, time scales, and mechanisms responsible forself-replication to study the underlying bifurcation hierarchiesthat organize the self-replication regime, to further examine thezero-pole cancellation phenomenon in the nonlocal eigenvalueproblem stability analysis, to extend the renormalization grouptechnique to establish the fully-nonlinear stability of pulses,and to develop extensions to systems with more than two lengthscales. Second, in the increasingly-important area of reductionmethods for large systems of chemical reactions with multipletime scales, the validity and accuracy of certain methods areanalyzed, with special focus on the computational singularperturbation method of Lam and Goussis. Third, the investigatoranalyzes the Oya-Vallochi model of subsurface bioremediation.Bioremediation is a process in which microorganisms, in thepresence of electron acceptors, degrade environmentally-harmfulorganic compounds. The investigator studies traveling waves ofbiomass activity and advection versus dispersion. Fourth, heconducts fundamental studies of nonspherical deformations of gasbubbles in Newtonian fluids. Finally, a challenging open problemconcerning the existence of self-similar, blow-up solutions ofthe nonlinear Schroedinger equation in spatial dimensions betweentwo and four is attempted. This project concerns mathematics for problems ofsignificant current interest in biology, chemistry, engineering,and physics, which exhibit both fast and slow dynamicalprocesses. First, with collaborators and a doctoral student, theinvestigator analyzes computational methods used to simulatelarge, complex systems of reactions in biochemistry, combustion,and air pollution engineering. These processes, such as theproduction of certain proteins, the burning of natural gas, andthe formation of nitrous oxides in the atmosphere, typicallyinvolve a few hundred species, each of which participates inseveral reactions, with the reaction times ranging fromnanoseconds to milliseconds, even to minutes. Methods that reducethe system complexity, while retaining a desired accuracy, arecritical for modeling these processes. The investigator aims toshow that there is a highly accurate method that can be used toimprove the accuracy of other widely-used methods, which areembedded in major computer codes. Second, the investigator and adoctoral student study mathematical models of bioremediation, inwhich microorganisms are used to degrade environmentally-harmfulorganic compounds. Mathematics provides an advantageous approachto determine important quantities, such as the wave speed withwhich the biologically-active zone propagates through a wet soilcolumn and how this speed depends on the many physicalparameters. Third, fundamental research is conducted on thedynamics of gas bubbles in water. Deformations of sphericalbubbles lead to oscillations on time scales much shorter thanthat on which the spherical mode itself oscillates, and the maingoal is to model the nonlinear transfer of energy between thespherical and nonspherical modes that can lead to bubblecavitation and the attendant production of underwater sound byturbine blades, for example. Finally, the investigator developsfurther theory for self-replicating chemical patterns and for aprototypical equation that governs nonlinear wave propagation.