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

项目申请：DMS - 0707445 PI: Matkowsky， Bernard j机构：Northwestern university题目：以异常扩散为模型的反应扩散系统中的模式形成和非线性动力学及其应用摘要提出的研究目标是建立具有异常扩散的反应扩散系统中的模式形成和非线性动力学理论。迄今为止所研究的大多数反应扩散系统的一个特征是扩散是正态的，即随机行走粒子的均方位移与时间的关系是线性的。然而，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，并且反应物的扩散是异常的。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，则比正常情况下快，称为超扩散。这两种类型的异常扩散已被认为在各种物理、化学、生物、地质和其他过程中发挥重要作用。对于反常扩散，反应扩散系统的偏微分方程被分数阶偏微分方程系统所取代，分数阶偏微分方程系统是由适当的连续时间随机游走模型在分子水平上推导出来的。分数阶方程等价于我们所研究的积分-微分方程组。提出的研究将包括一般的反应扩散系统与异常扩散，和具体的应用。一般反应扩散系统的研究将有助于阐明反常扩散条件下图案形成和非线性动力学的普遍机制。具体模型将包括著名的化学动力学模型，如Brusselator， Oregonator, Gray-Scott, Gierer-Meinhardt， Fitzhugh-Nagumo等模型，以及燃烧，聚合和受控药物输送模型。将考虑亚扩散和超扩散的影响。反应扩散系统在许多科学分支中无处不在，几十年来一直吸引着科学家、工程师和数学家的注意。由于图灵的突破性发现表明，化学反应物质混合物中的扩散可能导致空间均匀状态的不稳定性，从而形成时空模式，以及别洛乌索夫和Zhabotinskii发现了振荡化学反应，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模式的范例之一。螺旋波、空间规则、不同对称的固定模式（六边形、条纹等）以及化学湍流等迷人结构的形成，使反应扩散系统成为众多正在进行的研究的主题。然而，在许多系统中，输运过程不是通过正常扩散，而是通过异常扩散，异常扩散可能比正常扩散慢（亚扩散）或快（超扩散）。亚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的典型过程。虽然对异常扩散的许多方面进行了研究，但非线性动力学和模式形成方面的研究非常有限。这些初步研究表明了异常扩散的重要性和对其进行系统研究的必要性。pi将对异常扩散对反应扩散系统中图灵模式和空间非均匀振荡模式以及螺旋波形成的影响进行系统研究，具体应用于燃烧波、等温正面聚合波和受控药物递送。

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.