Current problems in nonlinear dynamics: Macroscopic modeling of microscopic interactions and instability of coherent structures

当前非线性动力学中的问题：微观相互作用和相干结构不稳定性的宏观建模当前许多复杂系统建模的挑战是建立涵盖从微观到宏观各个层面相互作用的综合描述。拟议研究的第一个主题就属于这一类，处理由活生物体组成的系统具有额外的复杂性。PI和合作者最近开发了一个流体动力学模型，描述了宏观水平上细菌菌落的动力学和生长。PI现在将研究细菌相互作用的微观动力学。“活的”相互作用粒子的数值模拟将被开发，在此基础上，粗粒化将被应用于宏观层面上的描述。研究将寻找一组碰撞规则，这些规则可以重现并解释最近在密集的细菌群落中观察到的集体行为。这一研究也将为未来生物集合动力学模型的发展奠定基础。提出研究的第二个主题涉及相干结构的稳定性。菌落的扭结-反扭结特征，弹性丝的局部变形，或者光纤中传播的超短脉冲，都是相干结构的例子。这些对象通常被描述为一个或多个模型偏微分方程的特殊解，现代分析技术已经发展到解决它们的稳定性问题。PI将把这些技术与数值计算和渐近展开相结合，以获得在非线性光学和非线性扩散的反应扩散方程中观察到的相干结构的稳定性和不稳定性结果。将在这样的研究中开发的工具和方法是通用的，足以应用于其他进化偏微分方程。数学建模在生命科学和物理科学中发挥着越来越重要的作用。特别是，基于我们对复杂系统的各种构建块如何相互作用的看法，并将这种看法转化为数学术语的模型，可以用来测试我们的理论，并确保我们的理解建立在坚实的基础上。这种方法面临的主要挑战是，尽管我们生活在宏观世界中，但我们的直觉往往在微观或积木层面上感觉更自在。所有的困难都在于弥合这两个世界之间的差距。这项工作的第一部分处理这样一个问题：目标是了解细菌之间的微观相互作用如何导致集体的宏观行为，正如在一些实验中观察到的那样。这种现象将在实验室培养细菌菌落的情况下进行建模，但类似的原理可以应用于生物膜的形成和动力学，这在工程和医学上有许多应用。工作的第二部分是关于分析非线性偏微分方程数学模型的一般方法。特别地，将研究一些与非线性光学和种群动力学有关的模型的解的稳定性。

Abstract: DMS-0405551 J Lega, University of ArizonaCurrent problems in nonlinear dynamics: Macroscopic modeling of microscopic interactions and instability of coherent structuresA current challenge in the modeling of many complex systems is todevelop comprehensive descriptions that cover all levels ofinteractions, from the microscopic to the macroscopic. The firsttopic of the proposed research falls into this category, with theadditional complexity of dealing with a system made of livingorganisms. The PI and a collaborator have recently developed ahydrodynamic model describing the dynamics and growth of bacterialcolonies at a macroscopic level. The PI will now investigate themicroscopic dynamics of interacting bacteria. Numericalsimulations of "live" interacting particles will be developed, onwhich coarse-graining will be applied to obtain a description atthe macroscopic level. The search will be for a set of collisionrules that can reproduce and explain the collective behaviors thathave recently been observed in dense colonies of bacteria. Thisstudy will also lay the ground for the future development ofkinetic models of collections of living organisms. The secondtopic of the proposed research concerns the stability of coherentstructures. The kink-anti-kink profile of a bacterial colony, thelocal deformations of an elastic filament, or the ultra-shortpulses propagating in an optical fiber, are all examples ofcoherent structures. Such objects are often described as specialsolutions of one or more model partial differential equations, andmodern analytical techniques have been developed to address thequestion of their stability. The PI will combine such techniqueswith numerical calculations and asymptotic expansions, in order toobtain stability and instability results for coherent structuresobserved in nonlinear optics and in reaction-diffusion equationswith nonlinear diffusion. The tools and methods that will bedeveloped in such a study are general enough to be applied toother evolutionary partial differential equations.Mathematical modeling is nowadays playing an important and growingrole in the life and physical sciences. In particular, models thatare based on our vision of how various building blocks of acomplex system interact with one another, and which translate thisvision into mathematical terms, can be used to test our theoriesand make sure our understanding is built on solid ground. The mainchallenge we face with such an approach is that our intuitionoften feels more at ease at the microscopic - or the buildingblock - level, even though we live in a macroscopic world. All ofthe difficulty lies in bridging the gap between these two worlds.The first part of this work deals with such a problem: the goal isto understand how microscopic interactions between bacteria maylead to collective, macroscopic behaviors, as observed in someexperiments. Such phenomena will be modeled in the case oflaboratory grown colonies of bacteria, but similar principlescould be applied to the formation and dynamics of biofilms, whichhave many engineering and medical applications. The second part ofthe work concerns general methods for the analysis of mathematicalmodels which involve nonlinear partial differential equations. Inparticular, the stability of solutions of some models relevant tononlinear optics and population dynamics will be investigated.