Dynamical systems and singular perturbation theory for multi-scale reaction-diffusion phenomena

国际和平研究所将对呈现多个时间尺度和多个长度尺度的科学问题进行研究。第一个项目集中在涉及波、锋、脉冲和斑点的化学模式的反应扩散模型上。半强相互作用理论和重整化群理论的理论框架将被用来理解多组分反应-扩散系统的散射和钉扎动力学，以及生物修复的反应-扩散-平流模型中的行波稳定性。在第二个项目中，PI将研究具有关于反应项的截断函数的二维R-D方程。这些截止点准确地模拟了低浓度区域。主要目标是分析这些截止点对传播锋面的速度和稳定性的影响。这将建立在PI开创性地使用几何去奇化来找到一维截止波速的解析公式的基础上，该解析公式与数值数据匹配得很好。在第三个项目中，PI将继续他的长期分析和开发精确的模型降阶方法。它们被用在大规模燃烧、化学和生化系统中，展示出多个时间尺度，以找到管理有效动力学的低维流形。这项新的研究将建立在PI对最常用的模型降阶方法(包括ILDM和CSP方法)的早期分析的基础上，并将通过开发和分析在存在扩散的情况下发现低维流形的方法来将该项目带到一个新的水平。在第四个项目中，将研究偏微分方程中的鸭子新现象，以及数学神经科学模型中的环面鸭子新现象。PI将对具有多时间尺度和多长度尺度的科学问题进行研究，目的是解释最近的实验，分析计算方法，发展新的数学理论。第一个项目以化学和生物系统中的图案和波为中心，目标是对条纹和斑点之间的相互作用进行建模。还将研究生物修复模型中的行波。生物修复是诱导微生物降解土壤中对环境有害的有机化合物的过程。PI已经建立了这些行波的基本属性，并将确定操作参数，如注入速度，从而在模型中实现稳定的生物修复。在第二个项目中，PI的目标是在存在截断的情况下在两个空间维度上模拟更具挑战性和更现实的锋面传播问题。确定截断如何改变锋面的速度和稳定性将在物理学中很有用，特别是对统计物理学中的多粒子系统。在第三个项目中，PI将专注于对燃烧科学、化学和生物化学中的大规模反应网络以及基因调控建模至关重要的模型简化方法。尽管计算方法的速度在不断提高，但对这些大规模反应系统的理解关键取决于对易于处理的低维简化模型的分析。在第四个项目中，鸭子是在不稳定的系统状态附近停留相对较长时间的解决方案。将被研究的新类型的鸭子在数学神经科学中对于理解基本状态之间的转换是重要的，例如强直的尖峰和爆发。所有上述计划中的研究都将产生更广泛的科学影响。将参与这些项目的博士生和博士后中有一半以上是女性，迄今为止，国际和平协会指导的20名博士生和博士后研究员也是如此。此外，计划中的项目是国家实验室和北约国家科学家合作的一部分。

The PI will conduct research on scientific problems exhibiting multiple time scales and multiple length scales. The first project centers on reaction-diffusion models for chemical patterns involving waves, fronts, pulses, and spots. The theoretical framework of semi-strong interaction theory and renormalization group theory, which the PI helped develop, will be used to understand the dynamics of scattering and pinning in multi-component reaction-diffusion systems, as well as the stability of traveling waves in a reaction-diffusion-advection model of bioremediation. In the second project, the PI will study R-D equations in two space dimensions with cut-off functions on the reaction terms. These cut-offs accurately model regions of low concentrations. The principal goal is to analyze the impact of these cut-offs on the speeds and stability of propagating fronts. This will build on the PI's pioneering use of geometric desingularization to find analytical formulas for wave speeds with cut-offs in 1-D, which match well the numerical data. In the third project, the PI will continue his long-term analysis and development of accurate model reduction methods. These are used in large-scale combustion, chemical, and biochemical systems exhibiting multiple time scales, to find low-dimensional manifolds that govern the effective dynamics. The new research will build on the PI's earlier analysis of the most commonly-used model reduction methods including the ILDM and CSP methods and will take this project to the next level by developing and analyzing methods that find low-dimensional manifolds in the presence of diffusion. In the fourth project, the new phenomena of canards in PDEs will be investigated, as will the new phenomenon of torus canards in mathematical neuroscience models.The PI will conduct research on scientific problems exhibiting multiple time scales and multiple length scales, with the goals of explaining recent experiments, of analyzing computational methods, and of developing new mathematical theory. The first project centers on patterns and waves in chemical and biological systems, with the goal of modeling the interactions between stripes and spots. Traveling waves in models of bioremediation will also be studied. Bioremediation is the process by which microorganisms are induced to degrade environmentally-harmful organic compounds in soil. The PI has established fundamental properties of these traveling waves, and will determine the operating parameters, such as injection rates, that lead to stable bioremediation within the models. In the second project, the PI aims to model the more challenging and realistic problem of front propagation in two space dimensions in the presence of cut-offs. Determination of how cut-offs change the speeds and stability of the fronts will be useful in physics, in particular for many-particle systems in statistical physics. In the third project, the PI will focus on model reduction methods that are critical for combustion science, large-scale reaction networks in chemistry and biochemistry, and in gene regulatory modeling. Despite the ever-advancing speed of computational methods, understanding of these large-scale reaction systems depends critically on the analysis of tractable low-dimensional reduced models. In the fourth project, canards are solutions that stay near unstable system states for relatively long periods of time. The new types of canards to be studied are important in mathematical neuroscience for understanding the transitions between fundamental states, such as tonic spiking and bursting. All of the above planned research will have broader scientific impacts. More than half of the PhD students and postdocs who will work on these projects are women, as is the case for the twenty PhD students and postdoctoral fellows whom the PI has supervised to date. Moreover, the planned projects are parts of collaborations involving scientists at national laboratories and in NATO countries.