Some classical integrable systems with topological solitons

This thesis is concerned with some low dimensional non-linear systems of partial differential equations and their solutions. The systems are all in the classical domain and aside from a version of one model in Appendix D, are continuous. To begin with we examine the field equations of motion derived from Hamiltonian and Lagrangian densities, respectively defining the (1 + 1)-dimensional Hyperbolic Heisenberg and Hyperbolic sigma models, where the metric on the target manifold is indefinite. The models are integrable in the sense that a suitable Lax pair exists, and admit solitonic solutions classifiable by an integer winding number. Such solutions are explicitly derived in both the static and time dependent cases where physical space X is the circle S(^1). The existence of travelling wave solutions of topological type is discussed for each model with X = S(^1) and X = R; explicit solutions are derived for the X = S(^1) case and it is shown for both the Heisenberg and sigma models, that no such travelling wave solutions exist if X is the real line. Nevertheless, time dependent solutions (not of travelling wave type) are possible in each case for X = R, some examples of which are derived explicitly. A further integrable system; the Hyperbolic 'Pivotal' model is proposed as a special case of a more general model on Hermitian symmetric spaces. Of particular interest is the fact that the Pivotal model interpolates between the previous two models. To begin with the integrability of the model is established via a Lax representation. Solutions analogous to some of those of the previous models are then derived and the interpolative limits examined with respect to the Heisenberg and sigma models. Conserved currents for the model are also briefly discussed. Finally, some conclusions and further possibilities are noted including a brief examination of a discrete version of the sigma model where the target manifold is positive definite. A Bogomol'nyi bound is shown to exist for the systems energy in terms of a well defined winding number

2-loop perturbative invariants of lens spaces and a test of Chern-Simons quantum field theory

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (p. 140-141).by Richard Stone.Ph.D

Dynamical systems associated with particle flow models : theory and numerical methods

A new class of integro - partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a novel limiting averaging method (inspired by techniques employed in the derivation of infinite-dimensional dynamical systems models) on the Newtonian equations of motion of a many-particle system incorporating widely used inelastic particle-particle force formulas. By using Taylor series expansions, these models can be approximated by a system of partial differential equations of the Navier-Stokes type. Solutions of the new models for granular flows down inclined planes and in vibrating beds are compared with known experimental and analytical results and good agreement is obtained. Theorems on the existence and uniqueness of a solution to the granular flow dynamical system are proved in the Faedo-Galerkin method framework. A class of one-dimensional models describing the dynamics of thin granular layers and some related problems of fluid mechanics was studied from the Liouville-Lax integrability theory point of view. The integrability structures for these dynamical systems were constructed using Cartan\u27s calculus of differential forms, Grassman algebras over jet-manifolds associated with the granular flow dynamical systems, the gradientholonomic algorithm and generalized Hamiltonian methods. By proving the exact integrability of the systems, the quasi-periodicity of the solutions was explained as well as the observed regularity of the numerical solutions. A numerical algorithm based on the idea of higher and lower modes separation in the theory of approximate inertial manifolds for dissipative evolutionary equations is developed in a finite-difference framework. The method is applied to the granular flow dynamical system. Numerical calculations show that this method has several advantages compared to standard finite-difference schemes. A numerical solution to the granular flow in a hopper is obtained using the finite difference scheme in curvilinear coordinates. By making coefficients in the governing equations functionally dependent on the gradient of the velocity field, we were able to model the influence of the stationary friction phenomena in solids and reproduce in this way experimentally observable results. Some analytical and numerical solutions to the dynamical system describing granular flows in vibrating beds are also presented. We found that even in the simplest case where we neglect the arching phenomena and surface waves, these solutions exhibit some of the typical features that have been observed in simulation and experimental studies of vibrating beds. The approximate analytical solutions to the governing system of equations were found to share several important features with actual granular flows. Using this approach we showed the existence of the typical dynamical structures of chaotic motion. By employing Melnikov theory the bifurcation parameter values were estimated analytically. The vortex solutions we obtained for the perturbed motion and the solutions corresponding to the vortex disintegration agree qualitatively with the dynamics obtained numerically

Safety criteria for aperiodic dynamical systems

The use of dynamical system models is commonplace in many areas of science and engineering. One is often interested in whether the attracting solutions in these models are robust to perturbations of the equations of motion. This question is extremely important in situations where it is undesirable to have a large response to perturbations for reasons of safety. An especially interesting case occurs when the perturbations are aperiodic and their exact form is unknown. Unfortunately, there is a lack of theory in the literature that deals with this situation. It would be extremely useful to have a practical technique that provides an upper bound on the size of the response for an arbitrary perturbation of given size. Estimates of this form would allow the simple determination of safety criteria that guarantee the response falls within some pre-specified safety limits. An excellent area of application for this technique would be engineering systems. Here one is frequently faced with the problem of obtaining safety criteria for systems that in operational use are subject to unknown, aperiodic perturbations. In this thesis I show that such safety criteria are easy to obtain by using the concept of persistence of hyperbolicity. This persistence result is well known in the theory of dynamical systems. The formulation I give is functional analytic in nature and this has the advantage that it is easy to generalise and is especially suited to the problem of unknown, aperiodic perturbations. The proof I give of the persistence theorem provides a technique for obtaining the safety estimates we want and the main part of this thesis is an investigation into how this can be practically done. The usefulness of the technique is illustrated through two example systems, both of which are forced oscillators. Firstly, I consider the case where the unforced oscillator has an asymptotically stable equilibrium. A good application of this is the problem of ship stability. The model is called the escape equation and has been argued to capture the relevant dynamics of a ship at sea. The problem is to find practical criteria that guarantee the ship does not capsize or go through large motions when there are external influences like wind and waves. I show how to provide good criteria which ensure a safe response when the external forcing is an arbitrary, bounded function of time. I also consider in some detail the phased-locked loop. This is a periodically forced oscillator which has an attracting periodic solution that is synchronised (or phase-locked) with the external forcing. It is interesting to consider the effect of small aperiodic variations in the external forcing. For hyperbolic solutions I show that the phase-locking persists and I give a method by which one can find an upperbound on the maximum size of the response

Nonlinear wave propagation in disordered media

We briefly review the state-of-the-art of research on nonlinear wave propagation in disordered media. The paper is intended to provide the non-specialist reader with a flavor of this active field of physics. Firstly, a general introduction to the subject is made. We describe the basic models and the ways to study disorder in connection with them. Secondly, analytical and numerical techniques suitable for this purpose are outlined. We summarize their features and comment on their respective advantages, drawbacks and applicability conditions. Thirdly, the Nonlinear Klein-Gordon and Schrbdinger equations are chosen as specific examples. We collect a number of results that are representative of the phenomena arising from the competition between nonlinearity and disorder. The review is concluded with some remarks on open questions, main current trends and possible further developments.This work has been supported in part by the C.I.C. y T. (Spain) under project MAT90-0S44. A S. was also supported by fellowships from the Universidad Complutense and the Ministerio de Educacion y Ciencia.Publicad

Generalized long-wave evolution equations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.Includes bibliographical references (p. 84-86).by Radica Šipčić.Ph.D

Asymptotics of wavelets and filters

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.Includes bibliographical references (p. 126-131).by Jianhong (Jackie) Shen.Ph.D