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

Rogue waves and solitons of the generalized modified nonlinear Schrodinger equations

Many applications of the classical nonlinear Schrodinger equations with cubic and power nonlinearity are seen in nonlinear optics, plasma physics, superconductivity, propagation of the electric field in optical fibers, self-focusing and collapse of Langmuir waves in plasma physics, to model deep water waves and freak waves in the ocean.Objectives: In this paper, the generalized form of the modified nonlinear Schrodinger equation is proposed with various nonlinearities.Methods: Bernoulli equation method, which is one of the ansatz-based methods, is considered to be obtained as the novel soliton solutions of the modified nonlinear Schrodinger equation with various nonlinearities.Results: With the view of the results, new improvements can happen for applications of the model.(c) 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved

Dark solitons in atomic Bose-Einstein condensates: from theory to experiments

This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose-Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher-dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.Comment: 82 pages, 13 figures. To appear in J. Phys. A: Math. Theor