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Yayın Standing waves for a generalized Davey-Stewartson system(IOP Publishing, 2006-10-27) Eden, Osman Alp; Erbay, SaadetIn this paper, we establish the existence of non-trivial solutions for a semi-linear elliptic partial differential equation with a non-local term. This result allows us to prove the existence of standing wave ( ground state) solutions for a generalized Davey-Stewartson system. A sharp upper bound is also obtained on the size of the initial values for which solutions exist globally.Yayın A note on the amplitude modulation of symmetric regularized long-wave equation with quartic nonlinearity(Springer, 2012-12) Demiray, HilmiWe study the amplitude modulation of a symmetric regularized long-wave equation with quartic nonlinearity through the use of the reductive perturbation method by introducing a new set of slow variables. The nonlinear Schrodinger (NLS) equation with seventh order nonlinearity is obtained as the evolution equation for the lowest order term in the perturbation expansion. It is also shown that the NLS equation with seventh order nonlinearity assumes an envelope type of solitary wave solution.Yayın Shapes and statistics of the rogue waves generated by chaotic ocean current(International Society of Offshore and Polar Engineers, 2016) Bayındır, CihanIn this study we discuss the shapes and statistics of the rogue (freak) waves emerging due to wave-current interactions. With this purpose, we use a simple governing equation which is a nonlinear Schrödinger equation (NLSE) extended by R. Smith (1976). This extended NLSE accounts for the effects of current gradient on the nonlinear dynamics of the ocean surface near blocking point. Using a split-step scheme we show that the extended NLSE of Smith is unstable against random chaotic perturbation in the current profile. Therefore the monochromatic wave field with unit amplitude turns into a chaotic sea state with many peaks. By comparing the numerical and analytical results, we show that rogue waves due to perturbations in the current profile are in the form of rational rogue wave solutions of the NLSE. We also discuss the effects of magnitude of the chaotic current profile perturbations on the statistics of the rogue wave generation at the ocean surface. The extension term in Smith's extended NLSE causes phase shifts and it does not change the total energy level of the wave field. Using the methodology adopted in this study, the dynamics of rogue wave occurrence on the ocean surface due to blocking effect of currents can be studied. This enhances the safety of the offshore operations and ocean travel.Yayın Higher order approximations in reductive perturbation method: Strongly dispersive waves(2005-08) Demiray, HilmiContribution of higher order terms in the perturbation expansion for the strongly dispersive ion-plasma waves is examined through the use of modified reductive perturbation method developed early by us. It is shown that the lowest order term in the expansion is governed by the nonlinear Schrödinger equation while the second-order term is governed by the linear Schrödinger equation. For the small wave number region a set of solution is presented for the evolution equations.Yayın Early detection of rogue waves by the wavelet transforms(Elsevier, 2016-01-08) Bayındır, CihanWe discuss the possible advantages of using the wavelet transform over the Fourier transform for the early detection of rogue waves. We show that the triangular wavelet spectra of the rogue waves can be detected at early stages of the development of rogue waves in a chaotic wave field. Compared to the Fourier spectra, the wavelet spectra are capable of detecting not only the emergence of a rogue wave but also its possible spatial (or temporal) location. Due to this fact, wavelet transform is also capable of predicting the characteristic distances between successive rogue waves. Therefore multiple simultaneous breaking of the successive rogue waves on ships or on the offshore structures can be predicted and avoided by smart designs and operations.Yayın Reducing a generalized Davey-Stewartson system to a non-local nonlinear Schrodinger equation(Pergamon-Elsevier Science Ltd, 2009-07-30) Eden, Osman Alp; Erbay, Saadet; Hacınlıyan, IrmaIn the present study, we consider a generalized (2 + 1) Davey-Stewartson (GDS) system consisting of a nonlinear Schrodinger (NLS) type equation for the complex amplitude of a short wave and two asymmetrically coupled linear wave equations for long waves propagating in an infinite elastic medium. We obtain integral representation of solutions to the coupled linear wave equations and reduce the GDS system to a NLS equation with non-local terms. Finally, we present localized solutions to the GDS system, decaying in both spatial coordinates, for a special choice of parameters by using the integral representation of solutions to the coupled linear wave equations.Yayın Analytical and numerical analysis of the dissipative kundu-eckhaus equation(Işık Üniversitesi, 2019-12-02) Yurtbak, Hazal; Bayındır, Cihan; Işık Üniversitesi, Fen Bilimleri Enstitüsü, İnşaat Mühendisliği Yüksek Lisans ProgramıIt is well-known that the Kundu-Eckhaus equation (KEE) is a nonlinear equation which belongs to nonlinear Schrödinger class and it is commonly used as a model to investigate the dynamics of diverse phenomena in many areas including but are not limited to hydrodynamics, fiber and nonlinear optics, plasmas and finance. However, the effects of dissipation on the dynamics of KEE have not been investigated so far. In this thesis, in order to address this open problem we propose the dissipative Kundu-Eckhaus equation (dKEE) and perform an analytical and numerical analysis of the dKEE. With this motivation, we derive a simple monochromatic wave solution to dKEE. Then, we propose a split step Fourier method (SSFM) for the numerical solution of the dKEE and we test the stability of the SSFM using the analytical solution derived as a benchmark problem. Observing the stability and the accuracy of the scheme, we first investigate the rogue wave dynamics of the dKEE using the SSFM. More specifically, we show that modulation instability (MI) turns the monochromatic wave field into a chaotic one, thus the appearance of rogue waves become obvious. We discuss the properties and characteristics of such rogue waves. Additionally, we depict the amplitude probability distribution functions (PDFs) and discuss the effects of diffusion, Raman and dissipation coeficient as well as the MI parameters on the probability of rogue wave occurrence. Secondly, we investigate the effects of dissipation on the self-localized solitons of the KEE. For this purpose, we propose a Petviashvili method (PM) to obtain the self-localized solitons of the KEE and analyze the effects of dissipation by time stepping of these solitons using the SSFM proposed for dKEE. It is known that, KEE admits stable single, two and N-soliton solutions for the no potential case. It has been recently found that, under the effect of photorefractive and saturable potentials, such solitons of the KEE become unstable. We show that the dissipation parameter can be used to stabilize the single, two and three solitons of the KEE which do not satisfy the necessary Vakhitov-Kolokolov condition for the soliton stability. With this aim, we present the power graphs as functions of soliton eigenvalue and as well as time. Additionally, we depict the soliton shapes for various times to show that they are preserved for time scales long enough for many engineering purposes.We comment on our findings and discuss the applicability and uses of our results. Additionally, we suggest possible directions for the near future research activities.Yayın Two-dimensional wave packets in an elastic solid with couple stresses(Pergamon-Elsevier Science Ltd, 2004-08) Babaoğlu, Ceni; Erbay, SaadetThe problem of (2+1) (two spatial and one temporal) dimensional wave propagation in a bulk medium composed of an elastic material with couple stresses is considered. The aim is to derive (2+1) non-linear model equations for the description of elastic waves in the far field. Using a multi-scale expansion of quasi-monochromatic wave solutions, it is shown that the modulation of waves is governed by a system of three non-linear evolution equations. These equations involve amplitudes of a short transverse wave, a long transverse wave and a long longitudinal wave, and will be called the "generalized Davey Stewartson equations". Under some restrictions on parameter values, the generalized Davey-Stewartson equations reduce to the Davey-Stewartson and to the non-linear Schrodinger equations. Finally, some special solutions involving sech-tanh-tanh and tanh-tanh-tanh type solitary wave solutions are presented.
