RUI: Existence and Stability of Coherent Structures with Applications to Elasticity

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Applications of elastic rod theory range from modeling thedynamics of DNA to the coiling of oceanic cables. The commonlyused descriptive equations for elastic rods dynamics are due toKirchhoff. However, the Kirchhoff model is complex and only fewclasses of explicit solutions are known and analyzed. In a seriesof papers, Goriely and coworkers obtained a reduced modelconsisting of amplitude equations for the dynamics of a twistedrod beyond the threshold of its first writhing instability. Solutions to these equations are known to correspond to fullythree-dimensional deformations of rods. The goals of this projectare twofold. Firstly, the investigator studies existence andstability properties for a large class of explicit solutions ofthe amplitude equations. Secondly, the methods developed to studythe stability of these solutions are extended to partialdifferential equations that are completely integrable or havecertain symmetry properties. More precisely, the first taskinvolves a systematic study of existence and stability of coherentstructures for amplitude equations. Existence is studied by meansof symmetry analysis, while stability is addressed using twodifferent tools: the Evans function and Hamiltonian techniques. The second task is a general study of the Evans function. Theinvestigator examines the relation between the Evans function andcomplete integrability and, in particular, is interested in theconnections with a particular notion of integrability, thePainleve Test. When integrability is not present, it is stillpossible for the equations to admit interesting Lie symmetries. The goal is to show that important information on the Evansfunction can be retrieved from these symmetries. Mechanics is a fascinating field of physics. Its mainpurpose is the study of interactions of particles in the presenceof forces such as gravity. In contrast to the dynamics ofparticles, the study of continuous media, such as fluids orelastic bodies, is challenging and not yet entirely wellunderstood. This is due to the fact that continuous media behavein numerous and complex ways. For example, it is difficult todescribe mathematically the twisting and bending of elastic rods,but approximate models do exist. The study of the dynamics ofelastic rods is a field of research that is interesting in itselfas it gives rise to beautiful and complex mathematical structures. Furthermore, understanding elastic rods is crucial forapplications in several domains of science. In biology, rodmodels are used to describe the coiling of different structuressuch as DNA. In engineering, elastic rods are used to study thebehavior of submarine cables. In this project, the investigatorfocuses on a model that has been proven to be successful indescribing the dynamics of rods: the Kirchhoff equations. Theseare differential equations, that is they are equations involvingderivatives. The solutions to these equations represent possiblebehaviors for elastic rods. The main purpose of the project is tostudy stability properties of elastic rod models. Stability is afundamental concept in physics: for example, in theory, it ispossible to make a pencil stand on its lead but, in practice,because that is such an unstable state, it cannot be done. In theexample just described, the study of stability is very simple andthere is no need for a mathematical analysis to prove or disprovethe stability of the system. The concept of stability carriesover to solutions of differential equations such as the Kirchhoffequations. In this context, stability takes an abstract form andits study often involves sophisticated mathematical tools. Thestudy of stability is of fundamental importance because onlystable solutions can be realized experimentally. Unstablesolutions, just like in the case of the pencil balancing on itspoint, although they exist in theory will not be observed. Hence,with stability studies, one can distinguish the solutions that canbe realized experimentally from the ones that cannot. Theinvestigator proposes new ways of investigating stabilitymathematically. The methods he develops are applied to elasticrods and other applications.