Δυναμική σκοτεινών σολιτονίων υπό την επίδραση διαταραχών (Master thesis)
In this study, we examine both numerically and analytically the dynamics of dark solitons under the influence of perturbations. Our aim is the implementation of the complete perturbation theory in order to approximate dark soliton solutions of the nonlinear Schrodinger (NLS) equation in the presence of specific higher-order effects. The basic idea leans on the adiabatic approximation of the perturbation theory for solitons. According to this approach the functional soliton shape remains unchanged, but a new slow scale is introduced, with which the evolution of the soliton parameters is studied. The adiabatic approximation permits to partition the problem into two regions, the inner region and the outer region. The inner region consists of the core soliton and the shelf. On the contrary, the outer region involves the boundary conditions at infinity, namely the continuous-wave background in which the dark soliton decays off. A boundary layer is introduced in order to match the two regions. As a result, a shelf develops and propagates around the soliton, which moves with similar velocity to the background velocity. It is shown that the dynamics of dark soliton depends on the evolution of the soliton parameters. The integration of the NLS equation, is an important quality, which arises from the inverse scattering transform (1ST), and gives us relations useful to study the evolution of these parameters. These are based on the conservation laws, which arise from the IST in an algorithmic way, and result in a system of differential equations. According to this system we describe the evolution of the soliton parameters for any perturbation. Moreover, the shelf has been confirmed and described asymptotically. For specific perturbations the analytical results which are derived by the system of ODEs are in accordance with the numerical results, for both constant and slowly evolving background. This analysis is applied to a wide range of physical problems, including linear and nonlinear dissipative perturbations.
|Institution and School/Department of submitter:||Πανεπιστήμιο Ιωαννίνων. Σχολή Θετικών Επιστημών. Τμήμα Μαθηματικών|
|Appears in Collections:||Διατριβές Μεταπτυχιακής Έρευνας (Masters)|
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|Μ.Ε. ΓΚΟΓΚΟΥ ΑΙΚΑΤΕΡΙΝΗ 2017.pdf||9.69 MB||Adobe PDF||View/Open|
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