Structure-preserving Finite Difference Methods for Linearly Damped Differential Equations

Differential equations (DEs) model a variety of physical phenomena in science and engineering. Many physical phenomena involve conservative or dissipative forces, which manifest themselves as qualitative properties of DEs that govern these phenomena. Since only a few and simplistic models are known to have exact solutions, approximate solution techniques, such as numerical integration, are used to reveal important insights about solution behavior and properties of these models. Numerical integrators generally result in undesirable quantitative and qualitative errors . Standard numerical integrators aim to reduce quantitative errors, whereas geometric (numerical) integrators aim to reduce or eliminate qualitative errors, as well, in order to improve the accuracy of numerical solutions. It is now widely recognized that geometric (or structure-preserving) integrators are advantageous compared to non-geometric integrators for DEs, especially for long time integration. Geometric integrators for conservative DEs have been proposed, analyzed, and investigated extensively in the literature. The motif of this thesis is to extend the idea of structure preservation to linearly damped DEs. More specifically, we develop, analyze, and implement geometric integrators for linearly damped ordinary and partial differential equations (ODEs and PDEs) that possess conformal invariants, which are qualitative properties that decay exponentially along any solution of the DE as the system evolves over time. In particular, we derive restrictions on the coefficient functions of exponential Runge-Kutta (ERK) numerical methods for preservation of certain conformal invariants of linearly damped ODEs. An important class of these methods is shown to preserve the damping rate of solutions of damped linear ODEs. Linearly stability and order of accuracy for some specific cases of ERK methods are investigated. Geometric integrators for PDEs are designed using structure-preserving ERK methods in space, time, or both. These integrators for PDEs are also shown to preserve additional structure in certain special cases. Numerical experiments illustrate higher order accuracy and structure preservation properties of various ERK based methods, demonstrating clear advantages over non-structure-preserving methods, as well as usefulness for solving a wide range of DEs

The algebraic structure of geometric flows in two dimensions

There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation associated to a supercontinual algebra with odd Cartan operator d/d \theta - \theta d/dt. Thus, taking the square root of the Cartan operator allows to connect the two distinct classes of geometric deformations of second and fourth order, respectively. The algebra is also used to construct formal solutions of the Calabi flow in terms of free fields by Backlund transformations, as for the Ricci flow. Some applications of the present framework to the general class of Robinson-Trautman metrics that describe spherical gravitational radiation in vacuum in four space-time dimensions are also discussed. Further iteration of the algorithm allows to construct an infinite hierarchy of higher order geometric flows, which are integrable in two dimensions and they admit immediate generalization to Kahler manifolds in all dimensions. These flows provide examples of more general deformations introduced by Calabi that preserve the Kahler class and minimize the quadratic curvature functional for extremal metrics.Comment: 54 page

Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background

We perform the study of the stability of the cosmological scalar field models, by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. In the KCC approach we describe the time evolution of the scalar field cosmologies in geometric terms, by performing a "second geometrization", by considering them as paths of a semispray. By introducing a non-linear connection and a Berwald type connection associated to the Friedmann and Klein-Gordon equations, five geometrical invariants can be constructed, with the second invariant giving the Jacobi stability of the cosmological model. We obtain all the relevant geometric quantities, and we formulate the condition of the Jacobi stability for scalar field cosmologies in the second order formalism. As an application of the developed methods we consider the Jacobi stability properties of the scalar fields with exponential and Higgs type potential. We find that the Universe dominated by a scalar field exponential potential is in Jacobi unstable state, while the cosmological evolution in the presence of Higgs fields has alternating stable and unstable phases. By using the standard first order formulation of the cosmological models as dynamical systems we have investigated the stability of the phantom quintessence and tachyonic scalar fields, by lifting the first order system to the tangent bundle. It turns out that in the presence of a power law potential both these models are Jacobi unstable during the entire cosmological evolution.Comment: 24 pages, 14 figures, accepted for publication in Advances in High Energy Physics, special issue "Dark Physics in the Early Universe

Lie systems: theory, generalisations, and applications

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure

Gauging of Geometric Actions and Integrable Hierarchies of KP Type

This work consist of two interrelated parts. First, we derive massive gauge-invariant generalizations of geometric actions on coadjoint orbits of arbitrary (infinite-dimensional) groups $G$ with central extensions, with gauge group $H$ being certain (infinite-dimensional) subgroup of $G$. We show that there exist generalized ``zero-curvature'' representation of the pertinent equations of motion on the coadjoint orbit. Second, in the special case of $G$ being Kac-Moody group the equations of motion of the underlying gauged WZNW geometric action are identified as additional-symmetry flows of generalized Drinfeld-Sokolov integrable hierarchies based on the loop algebra {\hat \cG}. For {\hat \cG} = {\hat {SL}}(M+R) the latter hiearchies are equivalent to a class of constrained (reduced) KP hierarchies called {\sl cKP}_{R,M}, which contain as special cases a series of well-known integrable systems (mKdV, AKNS, Fordy-Kulish, Yajima-Oikawa etc.). We describe in some detail the loop algebras of additional (non-isospectral) symmetries of {\sl cKP}_{R,M} hierarchies. Apart from gauged WZNW models, certain higher-dimensional nonlinear systems such as Davey-Stewartson and $N$-wave resonant systems are also identified as additional symmetry flows of {\sl cKP}_{R,M} hierarchies. Along the way we exhibit explicitly the interrelation between the Sato pseudo-differential operator formulation and the algebraic (generalized) Drinfeld-Sokolov formulation of {\sl cKP}_{R,M} hierarchies. Also we present the explicit derivation of the general Darboux-B\"acklund solutions of {\sl cKP}_{R,M} preserving their additional (non-isospectral) symmetries, which for R=1 contain among themselves solutions to the gauged $SL(M+1)/U(1)\times SL(M)$ WZNW field equations.Comment: LaTeX209, 47 page