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

University Of Arizona, Tucson AZ

Investigators

Abstract

Abstract: DMS-0405551 J Lega, University of Arizona Current problems in nonlinear dynamics: Macroscopic modeling of microscopic interactions and instability of coherent structures A current challenge in the modeling of many complex systems is to develop comprehensive descriptions that cover all levels of interactions, from the microscopic to the macroscopic. The first topic of the proposed research falls into this category, with the additional complexity of dealing with a system made of living organisms. The PI and a collaborator have recently developed a hydrodynamic model describing the dynamics and growth of bacterial colonies at a macroscopic level. The PI will now investigate the microscopic dynamics of interacting bacteria. Numerical simulations of "live" interacting particles will be developed, on which coarse-graining will be applied to obtain a description at the macroscopic level. The search will be for a set of collision rules that can reproduce and explain the collective behaviors that have recently been observed in dense colonies of bacteria. This study will also lay the ground for the future development of kinetic models of collections of living organisms. The second topic of the proposed research concerns the stability of coherent structures. The kink-anti-kink profile of a bacterial colony, the local deformations of an elastic filament, or the ultra-short pulses propagating in an optical fiber, are all examples of coherent structures. Such objects are often described as special solutions of one or more model partial differential equations, and modern analytical techniques have been developed to address the question of their stability. The PI will combine such techniques with numerical calculations and asymptotic expansions, in order to obtain stability and instability results for coherent structures observed in nonlinear optics and in reaction-diffusion equations with nonlinear diffusion. The tools and methods that will be developed in such a study are general enough to be applied to other evolutionary partial differential equations. Mathematical modeling is nowadays playing an important and growing role in the life and physical sciences. In particular, models that are based on our vision of how various building blocks of a complex system interact with one another, and which translate this vision into mathematical terms, can be used to test our theories and make sure our understanding is built on solid ground. The main challenge we face with such an approach is that our intuition often feels more at ease at the microscopic - or the building block - level, even though we live in a macroscopic world. All of the difficulty lies in bridging the gap between these two worlds. The first part of this work deals with such a problem: the goal is to understand how microscopic interactions between bacteria may lead to collective, macroscopic behaviors, as observed in some experiments. Such phenomena will be modeled in the case of laboratory grown colonies of bacteria, but similar principles could be applied to the formation and dynamics of biofilms, which have many engineering and medical applications. The second part of the work concerns general methods for the analysis of mathematical models which involve nonlinear partial differential equations. In particular, the stability of solutions of some models relevant to nonlinear optics and population dynamics will be investigated.