Geometric deformations of integrable systems and beyond
File(s)
Author(s)
Arnaudon, Alexis
Type
Thesis
Abstract
The results of this work were obtained within a single mathematical framework which consists of using the geometrical structure of a given problem not only to study its properties but to implement geometrically consistent deformations. The present interest in such geometric deformations is not only to learn more about the original problem but also to explore new directions, beginning from a common and well-understood starting point and continuing at the interface between several mathematical and physical topics. This approach lays down the foundation of this work upon which various ideas for deformations of several standard systems will be developed and analysed. The mathematical framework that we will be relying on throughout these investigations is the use of symmetries in dynamical systems described by Lie
groups which appear in many areas of mathematical physics and in particular in integrable systems, the second theme of this work. Indeed, the integrability of a dynamical system is often related to the existence of symmetries, but usually requires additional structures. The effect of the geometric deformations on the integrability of these systems will be one of the central questions that we will attempt to answer. To undertake such a program, we cannot restrict ourselves to the deformation of a single integrable system, but we must implement various ideas on several and different classic integrable equations in the aim of better characterising their effects and limitations. We will introduce noise, geometric dissipations and Sobolev norms in the classical integrable systems including the rigid body motion, the Toda lattice, the peakon system, partial differential equations such as KdV and NLS as well as the reduced Maxwell-Bloch equation.
groups which appear in many areas of mathematical physics and in particular in integrable systems, the second theme of this work. Indeed, the integrability of a dynamical system is often related to the existence of symmetries, but usually requires additional structures. The effect of the geometric deformations on the integrability of these systems will be one of the central questions that we will attempt to answer. To undertake such a program, we cannot restrict ourselves to the deformation of a single integrable system, but we must implement various ideas on several and different classic integrable equations in the aim of better characterising their effects and limitations. We will introduce noise, geometric dissipations and Sobolev norms in the classical integrable systems including the rigid body motion, the Toda lattice, the peakon system, partial differential equations such as KdV and NLS as well as the reduced Maxwell-Bloch equation.
Version
Open Access
Date Issued
2017-03
Date Awarded
2017-11
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Holm, Darryl
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)