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Öğe Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics(WORLD SCIENTIFIC PUBL CO PTE LTD, 2020) Yokus, Asif; Kaya, DoganThe traveling wave solutions of the combined Korteweg de Vries-modified Korteweg de Vries (cKdV-mKdV) equation and a complexly coupled KdV (CcKdV) equation are obtained by using the auto-Backlund Transformation Method (aBTM). To numerically approximate the exact solutions, the Finite Difference Method (FDM) is used. In addition, these exact traveling wave solutions and numerical solutions are compared by illustrating the tables and figures. Via the Fourier-von Neumann stability analysis, the stability of the FDM with the cKdV-mKdV equation is analyzed. The L-2 and L-infinity norm errors are given for the numerical solutions. The 2D and 3D figures of the obtained solutions to these equations are plotted.Öğe Exploring the influence of layer and neuron configurations on Boussinesq equation solutions via a bilinear neural network framework(Springer Science and Business Media B.V., 2024) Isah, Muhammad Abubakar; Yokus, Asif; Kaya, DoğanThis study examines the Boussinesq equation, which is a nonlinear partial differential equation used to describe long wave propagation in shallow water and has broader applications, including nonlinear lattice waves, vibrations in nonlinear strings, and ion sound waves in plasma. The Boussinesq equation provides an insight into the nonlinear long wave propagation behavior in shallow water by taking wave phase into account. Its versatility extends its utility beyond fluid dynamics to various physical phenomena. By providing specific activation functions in the ``2-3-1?? and ``2-5-1?? neural network models, respectively, the generalized lump solution and the precise analytical solutions are produced using the bilinear neural network approach. These analytical solutions, together with the related rogue waves, dark soliton, and bright soliton, are derived using symbolic computation. These findings fill in the gaps in the current research about the Boussinesq equation. The dynamical properties of these waves are displayed on three-dimensional, contour, density, and two-dimensional graphs. The response of the wave solution to different values of wave speed in relation to the wave phase it contains has been described with the help of wave intensity. In addition, the advantages and disadvantages of the layers used in the analytical technique to generate solutions have been discussed. The efficient techniques employed in this research are useful for studying the nonlinear differential equations in one-dimensional nonlinear lattice waves, vibrations in a nonlinear string, and ion sound waves in plasma.Öğe Nonlinear Dispersion Dynamics of Optical Solitons of Zoomeron Equation with New φ6Model Expansion Approach(L and H Scientific Publishing, LLC, 2024) Isah, Muhammad Abubakar; Yokus, AsifOne of the equations describing incognito evolution, the nonlinear Zoomeron equation, is studied in this work. In a variety of physical circumstances, including laser physics, fluid dynamics and nonlinear optics, solitons with particular properties arise and the Zoomeron equation is a single example of one such situation. The method of ?6-model expansion allows for the explicit retrieval of a wide range of solution types, including kink-type solitons, these solitons are also called topological solitons in the context of water waves, their velocities do not depend on the wave amplitude, others are bright, singular, periodic and combined singular soliton solutions. The outcomes of this research may improve the Zoomeron equation’s nonlinear dynamical features. The method proposes a practical and effective approach for solving a large class of nonlinear partial differential equations. The nonlinear dispersion behavior is analyzed for different values of the magnitude, which physically represents the wave velocity, from the parameters of the generated traveling wave solutions. Interesting graphs are employed to explain and highlight the dynamical aspects of the results, and all of the obtained results are put into the Zoomeron equation to show the accuracy of the results.Öğe Numerical and exact solutions for time fractional Burgers' equation(Int Scientific Research Publications, 2017) Yokus, Asif; Kaya, DoğanThe main purpose of this paper is to find an exact solution of the traveling wave equation of a nonlinear time fractional Burgers' equation using the expansion method and the Cole-Hopf transformation. For this purpose, a nonlinear time fractional Burgers' equation with the initial conditions considered. The finite difference method (FDM for short) which is based on the Caputo formula is used and some fractional differentials are introduced. The Burgers' equation is linearized by using the Cole-Hopf transformation for a stability of the FDM. It shows that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in L-2 and L-infinity. All of obtained results are discussed with an example of the Burgers' equation including numerical solutions for different situations of the fractional order and the behavior of potentials u is investigated with graphically. All the obtained numerical results in this study are presented in tables. We used the Mathematica software package in performing this numerical study. (C) 2017 All rights reserved.Öğe Optical solitons of the complex Ginzburg–Landau equation having dual power nonlinear form using φ6-model expansion approach(Mehmet Yavuz, 2023) Isah, Muhammad Abubakar; Yokus, AsifThis paper employs a novel ? 6 -model expansion approach to get dark, bright, periodic, dark-bright, and singular soliton solutions to the complex Ginzburg-Landau equation with dual power-law non linearity. The dual-power law found in photovoltaic materials is used to explain nonlinearity in the refractive index. The results of this paper may assist in comprehending some of the physical effects of various nonlinear physics models. For example, the hyperbolic sine arises in the calculation of the Roche limit and the gravitational potential of a cylinder, the hyperbolic tangent arises in the calculation of the magnetic moment and the rapidity of special relativity, and the hyperbolic cotangent arises in the Langevin function for magnetic polarization. Frequency values, one of the soliton’s internal dynamics, are used to examine the behavior of the traveling wave. Finally, some of the obtained solitons’ three-, two-dimensional, and contour graphs are plotted.Öğe Rogue waves and Stability Analysis of the new (2+1)-KdV Equation Based on Symbolic Computation Method via Hirota Bilinear Form(Institute of Electrical and Electronics Engineers Inc., 2023) Isah, Muhammad Abubakar; Yokus, AsifIn this article, we study the novel KdV model which makes a significant contribution to the understanding of a variety of nonlinear occurrences of ion-acoustic waves in plasma and acoustic waves in harmonic crystals. Hirota bilinearization will be used to carry out the task via the proper transformations. For various polynomial functions, several soliton solution types, specifically kink and rogue wave solutions, will be examined. We also present a kink wave and rogue wave interaction solution. The discovered solutions will be illustrated graphically. The existing work is widely used to report a variety of fascinating physical occurrences in the domains of shallow-water waves, ion-acoustic waves in plasma and acoustic waves in harmonic crystals.Öğe Role of Gilson-Pickering equation for the different types of soliton solutions: a nonlinear analysis(SPRINGER HEIDELBERG, 2020) Yokus, Asif; Durur, Hulya; Abro, Kashif Ali; Kaya, DoganIn this article, the soliton solutions of the Gilson-Pickering equation have been constructed using the sinh-Gordon function method (ShGFM) and (G '/G, 1/G)-expansion method, which are applied to obtain exact solutions of nonlinear partial differential equations. A solution function different from the solution function in the classical (G '/G, 1/G)-expansion method has been considered which are based on complex trigonometric, hyperbolic, and rational solutions. By invoking ShGFM and (G '/G, 1/G)-expansion methods, different traveling wave solutions have been investigated. For the sake of avoiding the complex calculations, the ready package program has been tackled. The comparative analysis of sinh-Gordon function and (G '/G, 1/G)-expansion methods has shown several differences and similarities. A comparative analysis of ShGFM and (G '/G, 1/G)-expansion methods assures that the (G '/G, 1/G)-expansion method has been found to be more intensive, powerful, reliable and effective method for the Gilson-Pickering equation. The graphical illustrations of two-, three-dimensional, and contour graphs have been depicted as well.Öğe Stability Analysis and Soliton Solutions of the Nonlinear Evolution Equation by Homoclinic Technique Based on Hirota Bilinear Form(Institute of Electrical and Electronics Engineers Inc., 2023) Yokus, Asif; Isah, Muhammad AbubakarThe novel KdV model plays a significant part in the discovery of a variety of nonlinear ion acoustic wave and harmonic crystal phenomena. The new homoclinic method based on the Hirota bilinear form is used to create the bilinear form of the new KdV equation and uncover numerous new exact solu tions. The stability of the solutions is studied in this article using the modulation instability. The results show novel mechanical structures and new properties for this evolution equation. The physical dynamics of the traveling wave solutions produced by the recently suggested homoclinic approach to reinforce the Hirota bilinear method are investigated, the obtained solutions are represented using 2?dimensional, 3?dimensional and contour plots. To guarantee their existence, all the solutions that have been found are inserted into the model. These results open up a new opportunity for us to thoroughly investigate the model. Numerous exciting physical occurrences in the fields of shallow-water waves, ion-acoustic waves in plasma, acoustic waves in harmonic crystal and other related phenomena are reported using the existing work on a regular basisÖğe A study on solitary wave solutions for the Zoomeron equation supported by two-dimensional dynamics(IOP Publishing Ltd, 2023) Duran, Serbay; Yokus, Asif; Kilinc, GulsenThis study emphasizes the importance of understanding natural phenomena through various observations and relating them to scientific studies. Nonlinear partial differential equations serve as fundamental tools for modeling these phenomena, with a focus on nonlinear evolution equations when involving time. This paper investigates the dynamics of the Zoomeron equation, highlighting not only a result derived from the KdV and Schrodinger equations but also its contribution to the modeling of Boomeron and Trappon solitons. Different analytical techniques yield various solutions for the Zoomeron equation, each offering unique insights. The behavior of the solutions generated in kink and singular soliton forms in various dynamics using the G'/G(2) -expansion method is investigated and compared under restrictive conditions, and the distribution of energy density is explained with the assistance of the gradient function. It is aimed at gaining a physically unique perspective by revealing the connections between the velocities of solitons produced for this equation and the effects of gradient flow directions. In addition, stability analysis for some of the solutions generated in the kink form is investigated. The structure of the paper includes method introduction, application, stability property, results and discussion, resulting in a unique perspective on understanding the physical dynamics of the Zoomeron model.
