Bright Soliton Solution of (1+1)-Dimensional Quantum System with Power-Law Dependent Nonlinearity

We study the nonlinear dynamics of (1+1)-dimensional quantum system in power-law dependent media based on the nonlinear Schrödinger equation (NLSE) incorporating power-law dependent nonlinearity, linear attenuation, self-steepening terms, and third-order dispersion term. The analytical bright soliton solution of this NLSE is derived via the F-expansion method. The key feature of the bright soliton solution is pictorially demonstrated, which together with typical analytical formulation of the soliton solution shows the applicability of our theoretical treatment

Bright solitary waves and non-equilibrium dynamics in atomic Bose-Einstein condensates

In this thesis we investigate the static properties and non-equilibrium dynamics of bright solitary waves in atomic Bose-Einstein condensates in the zero-temperature limit, and we investigate the non-equilibrium dynamics of a driven atomic Bose-Einstein condensate at finite temperature. Bright solitary waves in atomic Bose-Einstein condensates are non-dispersive and soliton-like matter-waves which could be used in future atom-interferometry experiments. Using the mean-field, Gross-Pitaevskii description, we propose an experimental scheme to generate pairs of bright solitary waves with controlled velocity and relative phase; this scheme could form an important part of a future atom interferometer, and we demonstrate that it can also be used to test the validity of the mean-field model of bright solitary waves. We also develop a method to quantitatively assess how soliton-like static, three-dimensional bright solitary waves are; this assessment is particularly relevant for the design of future experiments. In reality, the non-zero temperatures and highly non-equilibrium dynamics occurring in a bright solitary wave interferometer are likely to necessitate a theoretical description which explicitly accounts for the non-condensate fraction. We show that a second-order, number-conserving description offers a minimal self-consistent treatment of the relevant condensate -- non-condensate interactions at low temperatures and for moderate non-condensate fractions. We develop a method to obtain a fully-dynamical numerical solution to the integro-differential equations of motion of this description, and solve these equations for a driven, quasi-one-dimensional test system. We show that rapid non-condensate growth predicted by lower-order descriptions, and associated with linear dynamical instabilities, can be damped by the self-consistent treatment of interactions included in the second-order description

Molecular mechanisms of deformation of aligned polyethylene

This study investigated how aligned polyethylene transforms as a system. The aim was to identify the relative strength of symmetry and antisymmetry in the system. An empirical framework was developed based on the Lennard-Jones potential, which included three nested algorithms. These nested algorithms provided a context specific empirical study of aligned polyethylene. The first algorithm enabled the formation of a soliton in a classical structure, in order to minimise the energy of the system. This provided indicative results, which suggest that solitons may transfer load from one molecular chain to another. The correspondence between the formation of solitons and the restoring force on chain ends was also examined. The second algorithm included the classical formation of a soliton into a statistical structure, in order to examine the behaviour of the soliton. The algorithm showed that there are two distinct timescales associated with the structure of the soliton. The coarse-graining of the temporal structure of the system, followed by subsequent fine-graining, showed the splitting of the soliton into +/- π/2 twistons. The third algorithm included the behaviour of the soliton within a geometric construction consisting of hexagonal sites, to examine how spontaneous symmetry breaking and symmetry restoration may occur in a system that is at the critical point of orthorhombic and hexagonal symmetry. It was found that the system behaves neither as a pure solid crystal nor as a pure fluid. Therefore it may be predicted that the system has connection to a liquid crystal, which may inform novel efforts in the future to transform not just aligned polyethylene but also other similar materials.Open Acces

Defects of micropolar continua in Riemann-Cartan manifolds and its applications

We derive equations of motion and its solutions in the form of solitons from deformational energy functionals of a coupled system of microscopic and macroscopic deformations. Then criteria in constructing the chiral energy functional is specified to be included to obtain soliton-like solutions. We show various deformational measures, used in deriving the soliton solutions, can be written when both curvature and torsion are allowed, especially by means of microrotations and its derivatives. Classical compatibility conditions are re-interpreted leading to a universal process to derive a distinct set of compatibility conditions signifying a geometrical role of the Einstein tensor in Riemann-Cartan manifolds. Then we consider position-dependent axial configurations of the microrotations to construct intrinsically conserved currents. We show that associated charges can be written as integers under a finite energy requirement in connection with homotopic considerations. This further leads to a notion of topologically stable defects determined by invariant winding numbers for a given solution classification. Nematic liquid crystals are identified as a projective plane from a sphere hinted by the discrete symmetry in its directors. Order parameters are carefully defined to be used both in homotopic considerations and free energy expansion in the language of microcontinua. Micropolar continua are shown to be the general case of nematic liquid crystals in projective geometry, and in formulations of the order parameter, which is also the generalisation of the Higgs isovectors. Lastly we show that defect measures of pion fields description of the Skyrmions are related to the defect measures of the micropolar continua via correspondences between its underlying symmetries and compatibility conditions of vanishing curvature

The twistor description of integrable systems

The theory of twistors and the theory of integrable models have, for many years, developed independently of each other. However, in recent years it has been shown that there is considerable overlap between these two apparently disparate areas of mathematical physics. The aim of this thesis is twofold; firstly to show how many known integrable models may be given a natural geometrical/twistorial interpretation, and secondly to show how this leads to new integrable models, and in particular new higher dimensional models. After reviewing those elements of twistor theory that are needed in the thesis, a generalisation of the Yang-Mills self-duality equations is constructed. This is the framework into which many known examples of integrable models may be naturally fitted, and it also provides a simple way to construct higher dimensional generalisations of such models. Having constructed new examples of (2 + l)-dimensional integrable models, one of these is studied in more detail. Embedded within this system are the sine-Gordon and Non-Linear Schrodinger equations. Some solutions of this (2 + l)-dimensional integrable model are found using the 'Riemann Problem with Zeros' method, and these include the sohton solutions of the SG and NLS equations. The relation between this approach and one based the Atiyah-Ward ansatze is dicussed briefly. Scattering of localised structures in integrable models is very different from scattering in non-integrable models, and to illustrate this the scattering of vortices in a modified Abelian-Higgs model is considered. The scattering is studied, for small speeds, using the 'slow motion approximation' which involves the calculation of a moduli space metric. This metric is found for a general TV-lump vortex configuration. Various examples of scattering processes are discussed, and compared with scattering in an integrable model. Finally this geometrical approach is compared with other approaches to the study of integrable systems, such as the Hirota method. The thesis closes with some suggestions for how the KP equation may be fitted into this geometrical/twistorial scheme

Local wellposedness and global regularity results for biharmonic wave maps

This thesis is concerned with biharmonic wave maps, i.e. a bi-harmonic version of the wave maps equation, which is a Hamiltonian equation for a higher order energy functional and arises variationally from an elastic action functional for a manifold valued map.$\\[1pt]$ In the first part we present local and global results from energy estimates for biharmonic wave maps into compact, embedded target manifolds. This includes local wellposedness in high regularity and global regularity in subcritical dimension $n = 1, 2$. The results rely on the use of careful a priori energy estimates, compactness arguments in weak topologies and sharp Sobolev embeddings combined with energy conservation in the proof of global regularity.$\\[1pt]$ In part two, we extend these results to global regularity in dimension $n \geq 3$ for biharmonic wave maps into spheres and initial data of small size in a scale invariant Besov norm. This follows from a small data global wellposedness and persistence of regularity result for more general systems of biharmonic wave equations with non-generic nonlinearity. In contrast to part one, the arguments in part two of the thesis rely on the analysis of bilinear frequency interactions based on Fourier restriction methods and Strichartz estimates.$\\[1pt]$ The results in both parts of the thesis fundamentally depend on the non-generic form of the nonlinearity that is introduced by our biharmonic model problem

Modelling and Design of Advanced High Speed Vertical Cavity Semiconductor Lasers

Vertical-cavity surface-emitting laser (VCSEL) constructions capable of direct modulation at bit rates in excess of 40 GBit/s have attracted considerable attention for future high speed long- and medium-haul networks. The two main approaches to realising this goal are, firstly, the improvement in the direct current modulation laser performance, with 40 GBit/s direct modulation having been demonstrated recently, and, secondly, using advanced modulation schemes. These, in turn, fall into two major categories: firstly, modulation of the photon lifetime in the cavity as an alternative to current modulation, and, secondly, current modulation enhanced by photon-photon resonance in a specialised laser structure (e.g. using an external cavity [1], or a laser array [2]). Theoretical models describing both of these solutions have been developed, but appear to have certain limitations which will be discussed later in the thesis, and no systematic analysis and comparison of modulation properties of advanced modulation scheme had been performed, to the best of my knowledge. This was the purpose of my PhD project. In order to understand the performance of the photon lifetime modulation for Compound Vertical Cavity Surface Emitting Semiconductor Lasers more accurately, a model involving careful analysis of both amplitude and frequency (phase) of laser emission, as well as the spectrally selective nature of the laser cavity, is required. We have developed such a model and used it to describe the laser operation and predict the performance beyond current experimental conditions in both large and small signal modulation regimes for the first time according to our knowledge. Finally, we studied the alternative method of ultrafast modulation of VCSELs, consisting of current modulation enhanced by photon-photon resonance. The analysis concentrates on the version of the method involving an in-plane integrated extended cavity. A new model is developed to overcome the limitations of existing models and to allow better understanding of the dynamic of the in-plane laser cavity