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

College Of Charleston, Charleston SC

Investigators

Abstract

Applications of elastic rod theory range from modeling the dynamics of DNA to the coiling of oceanic cables. The commonly used descriptive equations for elastic rods dynamics are due to Kirchhoff. However, the Kirchhoff model is complex and only few classes of explicit solutions are known and analyzed. In a series of papers, Goriely and coworkers obtained a reduced model consisting of amplitude equations for the dynamics of a twisted rod beyond the threshold of its first writhing instability. Solutions to these equations are known to correspond to fully three-dimensional deformations of rods. The goals of this project are twofold. Firstly, the investigator studies existence and stability properties for a large class of explicit solutions of the amplitude equations. Secondly, the methods developed to study the stability of these solutions are extended to partial differential equations that are completely integrable or have certain symmetry properties. More precisely, the first task involves a systematic study of existence and stability of coherent structures for amplitude equations. Existence is studied by means of symmetry analysis, while stability is addressed using two different tools: the Evans function and Hamiltonian techniques. The second task is a general study of the Evans function. The investigator examines the relation between the Evans function and complete integrability and, in particular, is interested in the connections with a particular notion of integrability, the Painleve Test. When integrability is not present, it is still possible for the equations to admit interesting Lie symmetries. The goal is to show that important information on the Evans function can be retrieved from these symmetries. Mechanics is a fascinating field of physics. Its main purpose is the study of interactions of particles in the presence of forces such as gravity. In contrast to the dynamics of particles, the study of continuous media, such as fluids or elastic bodies, is challenging and not yet entirely well understood. This is due to the fact that continuous media behave in numerous and complex ways. For example, it is difficult to describe mathematically the twisting and bending of elastic rods, but approximate models do exist. The study of the dynamics of elastic rods is a field of research that is interesting in itself as it gives rise to beautiful and complex mathematical structures. Furthermore, understanding elastic rods is crucial for applications in several domains of science. In biology, rod models are used to describe the coiling of different structures such as DNA. In engineering, elastic rods are used to study the behavior of submarine cables. In this project, the investigator focuses on a model that has been proven to be successful in describing the dynamics of rods: the Kirchhoff equations. These are differential equations, that is they are equations involving derivatives. The solutions to these equations represent possible behaviors for elastic rods. The main purpose of the project is to study stability properties of elastic rod models. Stability is a fundamental concept in physics: for example, in theory, it is possible to make a pencil stand on its lead but, in practice, because that is such an unstable state, it cannot be done. In the example just described, the study of stability is very simple and there is no need for a mathematical analysis to prove or disprove the stability of the system. The concept of stability carries over to solutions of differential equations such as the Kirchhoff equations. In this context, stability takes an abstract form and its study often involves sophisticated mathematical tools. The study of stability is of fundamental importance because only stable solutions can be realized experimentally. Unstable solutions, just like in the case of the pencil balancing on its point, although they exist in theory will not be observed. Hence, with stability studies, one can distinguish the solutions that can be realized experimentally from the ones that cannot. The investigator proposes new ways of investigating stability mathematically. The methods he develops are applied to elastic rods and other applications.