Pattern Formation and Nonlinear Dynamics in Reaction-Diffusion Systems Modeled by Anomalous Diffusion, and Applications

建议：DMS-0707445 PI：Matkowsky，Bernard J研究所：西北大学标题：由反常扩散建模的反应-扩散系统中的图案形成和非线性动力学，及其应用摘要本研究的目的是发展具有反常扩散的反应-扩散系统中的图案形成和非线性动力学理论。迄今为止已研究的大多数反应扩散系统的一个特征是扩散是正常的，即随机行走的粒子的均方位移对时间的依赖是线性的，但在某些情况下观察到更一般的依赖关系，其中均方位移表现为时间的幂函数，而反应物的扩散是反常的。如果指数小于1，则扩散过程比正常扩散慢，称为次扩散。相反，如果指数大于1，则比正常情况下的速度快，称为超扩散。这两种类型的异常扩散都被认为在各种物理、化学、生物、地质和其他过程中发挥着重要作用。对于反常扩散，用分子水平上适当的连续时间随机游动模型导出的分数阶偏微分方程组代替反应扩散偏微分方程组。分数阶方程等价于我们所研究的积分-微分方程组。建议的研究既涉及具有反常扩散的一般反应扩散系统，也涉及具体应用。对一般反应扩散系统的研究将有助于阐明反常扩散条件下斑图形成和非线性动力学的普遍机制。具体的模型将包括著名的化学动力学模型，如布鲁塞尔模型、俄勒冈州模型、格雷-斯科特模型、吉耶尔-迈因哈特模型、菲茨休-纳古莫模型和其他模型，以及燃烧、聚合和受控药物输送模型。反应扩散系统在许多科学分支中普遍存在，几十年来一直吸引着科学家、工程师和数学家的注意。自从图灵的开创性发现表明化学反应物种的混合物中的扩散会导致空间均匀状态的不稳定导致时空图案的形成，以及别洛索夫和扎博廷斯基发现了振荡化学反应以来，反应扩散系统已经成为远离热力学平衡的系统中形成时空图案的范例之一。螺旋波、空间规则的、具有不同对称性(六边形、条纹等)的静止图案等迷人结构的形成以及化学湍流，使反应扩散系统成为许多正在进行的研究的主题。然而，在许多系统中，传输过程不是通过正常扩散，而是通过反常扩散，反常扩散可能比正常扩散慢(次扩散)或快(超扩散)。二次扩散通常发生在生物凝胶、多孔介质和聚合物中，而超扩散是等离子体、半导体、表面反应和许多其他过程中的典型过程。虽然已经研究了反常扩散的许多方面，但非线性动力学和模式形成方面的研究成果非常有限。这些初步研究表明了异常扩散的重要性和系统研究的必要性。PIS将系统地研究异常扩散对反应-扩散系统中图灵图案和空间不均匀振荡图案以及螺旋波的形成的影响，并具体应用于燃烧波、等温前沿聚合波和受控药物输送。

Proposal: DMS - 0707445 PI: Matkowsky, Bernard JInstitution: Northwestern UniversityTitle: Pattern Formation and Nonlinear Dynamics in Reaction-Diffusion Systems Modeled by Anomalous Diffusion, and Applications AbstractThe goal of the proposed study is to develop a theory of pattern formation and nonlinear dynamics in reaction-diffusion systems with anomalous diffusion. A characteristic feature of most of the reaction-diffusion systems that have been studied to date is that diffusion is normal, i.e., the dependence of the mean square displacement of a randomly walking particle on time is linear.In some cases, however, a more general dependence is observed, in which the mean square displacement behaves as a power function of time, and the diffusion of the reactants is anomalous. If the exponent is less than one the diffusion process is slower than normal diffusion and is called subdiffusion. In contrast, if the exponent is greater than one it is faster than normal and is called superdiffusion. Both types of anomalous diffusion have been recognized to play important roles in various physical, chemical, biological, geological, and other processes. For anomalous diffusion the reaction-diffusion system of partial differential equations is replaced by a system of fractional partial differential equations which have been derived from appropriate continuous time random walk models at the molecular level. The fractional equations are equivalent to systems of integro-differential equations which are the objects of our investigation. The proposed studies will involve both general reaction-diffusion systems with anomalous diffusion, and specific applications. Investigations of general reaction-diffusion systems will help elucidate the universal mechanisms of pattern formation and nonlinear dynamics under anomalous diffusion conditions. The specific models will include the famous chemical dynamics models such as the Brusselator, Oregonator, Gray-Scott, Gierer-Meinhardt, Fitzhugh-Nagumo, and other models, as well as models for combustion, polymerization, and controlled drug delivery. The effects of both sub- and superdiffusion will be considered.Reaction-diffusion systems are ubiquitous in many branches of science and have been attracting the attention of scientists, engineers and mathematicians for decades. Since the ground-breaking discoveries of Turing who showed that diffusion in a mixture of chemically reacting species could cause instability of a spatially uniform state leading to the formation of spatio-temporal patterns, and Belousov and Zhabotinskii who discovered oscillating chemical reactions, reaction-diffusion systems have become one of the paradigms for the formation of spatio-temporal patterns in systems far from thermodynamic equilibrium. The formation of such fascinating structures as spiral waves, spatially-regular, stationary patterns with different symmetries (hexagonal, stripe,etc.) as well as chemical turbulence have made reaction-diffusion systems the subject of numerous ongoing investigations. In many systems, however, the transport process is not via normal diffusion but rather via anomalous diffusion which can be slower (subdiffusion) or faster (superdiffusion) than normal diffusion. Subdiffusion often occurs in biogels, porous media and polymers, while superdiffusion is typical of some processes in plasmas, semiconductors, surface reactions and many others. Although many aspects of anomalous diffusion have been studied, nonlinear dynamic and pattern formation aspects were the subject of only a very limited number of works. These preliminary studies demonstrate the significance of anomalous diffusion and the necessity for its systematic investigation. The PIs will conduct systematic studies of the effect of anomalous diffusion on the formation of Turing patterns and spatially nonuniform oscillating patterns as well as spiral waves in reaction-diffusion systems, with specific applications to combustion waves, isothermal frontal polymerization waves, and controlled drug delivery.