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Yayın Analytical and numerical aspects of the dissipative nonlinear Schrödinger equation(Işık University Press, 2016-02-15) Bayındır, CihanIn this paper various analytical and numerical aspects of the dissipative nonlinear Schrodinger equation (d-NLS equation) are discussed. Decaying solitary wave type solutions derived by Demiray is reviewed and a new approximate solitary wave type solution of the d-NLS equation is introduced in order to make comparisons. Also a split-step Fourier scheme is proposed for numerical solution of the d-NLS equation and the analytical solutions are compared with the numerical results.Yayın Rogue wavefunctions due to noisy quantum tunneling potentials(Işık University Press, 2017-02-02) Bayındır, CihanIn this paper, we study the effects of white-noised potentials on nonlinear quantum tunneling. We use a split-step scheme to numerically solve the nonlinear Schrodinger equation (NLSE) with a tunneling potential. We consider three different types of potentials, namely; the single rectangular barrier, double rectangular barrier, and triangular barrier. For all these three cases, we show that white-noise given to potentials do not trigger modulation instability for tunneling of the sech type soliton solutions of the NLSE. However, white-noised potentials trigger modulation instability for tunneling of the sinusoidal wavefunctions; thus, such a wavefield turns into a chaotic one with many apparent peaks. We argue that peaks of such a field may be in the form of rational rogue wave solutions of the NLSE. Our results can be used to examine the effects of noise on quantum tunneling. Since a rogue wavefunction means a higher probability of the tunneling particle to be at a given (x,t) coordinate, our results may also be used for developing the quantum science and technology with many possible applications including but are not limited to increasing the resolution and efficiency of scanning tunneling microscopes, enhancing proton tunneling for DNA mutation and enhancing superconducting properties of junctions.Yayın Compressive split-step Fourier method(Işık University Press, 2015-02-09) Bayındır, CihanIn this paper an approach for decreasing the computational effort required for the split-step Fourier method (SSFM) is introduced. It is shown that using the sparsity property of the simulated signals, the compressive sampling algorithm can be used as a very efficient tool for the split-step spectral simulations of various phenomena which can be modeled by using differential equations. The proposed method depends on the idea of using a smaller number of spectral components compared to the classical split-step Fourier method with a high number of components. After performing the time integration with a smaller number of spectral components and using the compressive sampling technique with l(1) minimization, it is shown that the sparse signal can be reconstructed with a significantly better efficiency compared to the classical split-step Fourier method. Proposed method can be named as compressive split-step Fourier method (CSSFM). For testing of the proposed method the Nonlinear Schrodinger Equation and its one-soliton and two-soliton solutions are considered.Yayın Modeling the effects of soil improvement on train induced random ground-borne vibrations(Isik University, 2025-05-01) Bayındır, Cihan; Kesten, Ali Sercan; Etminan, EhsanGround-borne vibrations by railway trains are generated at the rail-wheel interface due to the passage of wheels and due to irregularities of wheels and tracks. These vibrations need to be predicted and controlled during the design and service of the railway for the safety and serviceability of the railway to avoid possible vibrationinduced problems such as settlement and differential settlement due to their compaction effect, liquefaction, and discomfort of people. While such railway vibrations are modeled by different techniques, only a few studies do exist to analyze them in the case of soilimproved conditions. In this study, we propose a mathematical framework to study the effects of soil improvement on the ground-borne vibrations induced by railway trains. We use an experimentally calibrated model that utilizes the evolutionary random process approach to model the time-varying transfer functions between the axles of the train and the fixed observation point. The railway is modeled as a Winkler foundation with rail pads and corresponding transfer functions are used. The target area of this study is the Emin¨on¨u-Alibeyk¨oy Tramway Line in ˙Istanbul, which is under construction. Due to poor soil conditions at the specific stations along the proposed tramway route, soil improvement by the application of geo-synthetics is performed at the site and taken into account in our model. The improvement in soil conditions is modeled as increased vertical soil stiffness in the Winkler foundation of the evolutionary random process model. To model the various tramway loading conditions, both the 5-axle and 6-axle tramway configurations with non-uniform axle spacing are considered. We show that by increasing the vertical soil stiffness ksb, the vibration velocity and acceleration levels can be reduced significantly. By implementing the model proposed, we present the reduction of the vibration velocity and acceleration levels as the functions of soil improvement parameters and discuss our findings and the applicability of the model.
