Browsing by Author "Barashenkov, Igor"

Now showing 1 - 6 of 6
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    Geometrical and nonperturbative aspects of low dimensional field theories
    (2000) Murugan, Jeffrey; Barashenkov, Igor
    We present a collection of results on solitons in low-dimensional classical field theory. We begin by reviewing the geometrical setting of he nonlinear ơ-model and demonstrate the integrability of the theory in two-dimensions on a symmetric target manifold. After reviewing the construction of soliton solutions in the 0(3) ơ-model we consider a class of gauged nonlinear ơ-models on two-dimensional axially-symmetric target spaces. We show that, for a certain choice of self-interaction, these models are all self-dual and analyze the resulting Bogomol'nyi equations in the BPS limit using techniques from dynamical systems theory. Our analysis is then extended to topologically massive gauge fields. We conclude with a deviation into exploring links between four-dimensional self-dual Yang-Mills equations and various lower-dimensional field theories. In particular, we show that at the level of equations of motion, the Euclidean Yang-Mills equations in light-cone coordinates reduce to the two-dimensional nonlinear ơ-model.
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    Localised solutions of the parametrically driven Ginzburg-Landau and nonlinear Schrӧdinger equations
    (2003) Cross, Simon; Barashenkov, Igor
    This thesis deals with localised solutions of the parametrically driven Ginzburg-Landau equation and its nonlinear Schrӧdinger limit. We begin with a detailed analysis of the Faraday Resonance experiment, in which the driven complex Ginzburg-Landau equation (CGLE) arises, and an examination of how the CGLE appears as the amplitude equation for the modes excited near a Hopf bifurcation.
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    Solitons and radiation in nonintegrable systems
    (2007) Oxtoby, Oliver Francis; Barashenkov, Igor
    Word processed copy. Includes bibliographical references (p. [171]-182).
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    Solitons, spatiotemporal chaos and synchronization in arrays of damped driven nonlinear oscillators
    (2008) Robinson, William Michael Lewin; Barashenkov, Igor; Alexeeva, Nora
    Simulating the homogeneous chain, we demonstrate the existance of periodically oscillating solitons, the period-doubling sequence of transitions to temporal chaos and the degeneration into spatiotemporal chaos for larger driving strengths. Next, we explore the effect of introducing weak disorder in the simplest form, a single impurity, into the chain. We describe how a long impurity can induce a pinned soliton to form and prevent spatiotemporal chaos, synchronizing the oscillators in the chain. Lastly we study chains with several impurities in various configurations.
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    Solitons,spatiotemporal choas and synchronization in arrays of damped driven nonlinear oscillators
    (2008) Robinson,WML; Barashenkov, Igor; Alexeeva, Nora
    We explore periodic and chaotic motion in a soliton-bearing chain of nonlinear oscillators and study the effect of disorder on the spatiotemporal chaos. The chain we consider consists of torsionally coupled, damped, parametrically driven pendulums. We show that the amplitudes of the pendulums satisfy a system of equations in slow time, the "nonlinear Schrodinger (NLS) oscillators". The evolution of the chain of NLS oscillators is simulated numerically. Simulating the homogeneous chain, we demonstrate the existence of periodically oscillating solitons, the period-doubling sequence of transitions to temporal chaos and the degeneration into spatiotemporal chaos for larger driving strengths. ~ext, we explore the effect of introducing weak disorder in the simplest form, a single impurity, into the chain. We describe how a long impurity can induce a pinned soliton to form and prevent spatiotcmporal chaos, synchronizing the oscillators in the chain. Lastly, we study chains with several impurities in various configurations. Cases considered include chains with multiple impurities of different strengths, equal equidistant impurities (both long and short), and equal impurities which are positioned randomly along the chain. vVe show that all the previously reported features of the continuous NLS equation are reproduced in the dynamics of the discrete chain. Our results indicate that the dominant structures of the spatiotemporal chaos are unstable solitons. We show how spatiotemporally chaotic motion can be suppressed by disordering the chain through the inclusion of one or more impurities, synchronizing the oscillators. A comprehensive analysis of chains with identical equidistant long impu- rities reveals an unexpectedly complex relationship between the number and strength of impurities and the dynamics observed. As a result of disordering the positions of impurities, we find that oscillatiug solitons form between impurities when the gaps are large enough. The oscillations may be periodic or chaotic. Equidistant short impurities may also stabilize the chain. We conclude that for a 8et of irnpuritie8 to prevent spatiotemporal chaos from emerging in the array, the intervals between the impurities should be sufficiently small, and the stre11gth of the impuritic8 should be sufficiently large. The required number and strength of impurities depends on the oscillator parameters and the initial condition of the chain.
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    Spectral continuation study of the temporally periodic solitons of the damped-driven nonlinear Schrödinger equations
    (2012) Lee-Thorpe, James; Barashenkov, Igor
    In this thesis we develop and employ a spectral continuation algorithm, implemented in AUTO, to study the temporally periodic spatially localised soliton solutions of the driven, damped nonlinear Schrödinger equations, both in the case of parametric driving and direct driving. We hope that this study is of interest not only in the context of the nonlinear Schrödinger equations but also separately as a study of an efficient numerical algorithm for continuing (path-following) solutions to general two-dimensional periodic soliton bearing PDEs.