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

摘要：DMS-0405551 J Lega，亚利桑那大学非线性动力学中的当前问题：微观相互作用的宏观建模和相干结构的不稳定性许多复杂系统建模中的当前挑战是开发涵盖从微观到宏观的所有层次的相互作用的全面描述。拟议研究的第一个主题福尔斯就属于这一范畴，但要处理由活生物体组成的系统，这又增加了一个复杂性。PI和一个合作者最近开发了一个流体动力学模型，在宏观水平上描述了细菌菌落的动力学和生长。PI现在将研究相互作用的细菌的微观动力学。将开发“活的”相互作用粒子的数值模拟，在其上将应用粗粒化以获得宏观水平的描述。这项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。这项研究也将奠定了基础，为未来的发展ofkinetic模型的收集活生物体。建议研究的第二个主题涉及相干结构的稳定性。细菌菌落的扭结-反扭结轮廓、弹性细丝的局部变形或在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解，现代分析技术已经发展到解决其稳定性问题。PI将联合收割机与数值计算和渐近展开相结合，以获得在非线性光学和具有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。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愿景的模型，以及将这种愿景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，我们的直觉通常在微观或积木层面上感觉更轻松，即使我们生活在宏观世界中。所有的困难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，宏观的行为，正如在一些实验中观察到的那样。这种现象将在实验室生长的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，这具有许多工程和医学应用。第二部分的工作涉及的一般方法的分析prosticalmodels，其中涉及非线性偏微分方程。特别地，将研究与非线性光学和种群动力学有关的一些模型的解的稳定性。

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.