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

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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.