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The aim of this study is to peruse the dispersive Schrödinger–Hirota equation (SH equation) with parabolic law linearity to get optical soliton solutions by utilizing the generalized Kudryashov method (GKM). Firstly, we introduced the solution algorithm of GKM. Then, we substituted the traveling wave transform to dispersive SH equation with parabolic law and gained nonlinear ordinary differential equation (NODE) form of real and imaginary parts. We successfully applied GKM to the NODE form of dispersive SH equation and obtained the appropriate solution sets. Using these sets, the solution of differential equation and necessary transformations, we created the analytical solution functions of dispersive SH equation and demonstrated the graphical simulations. Optical soliton solutions were obtained, graphical representations were made and interpreted, at the same time, it was observed that some parameters had an effect on the soliton behavior. The parabolic law form of the examined model has not been studied, and the obtained results have not been reported. [ABSTRACT FROM AUTHOR]

In this study, the analytical solution of the Hirota equation by using the rational sine–Gordon expansion method (RSGEM) has been investigated. For this purpose, the mentioned model is converted into an ordinary differential equation by using a wave transformation. After the balancing procedure, a set of equation system is found. The solutions of the equation system give coefficients both in the considered solution form and in the model. The proposed technique gives new analytical solutions including the dark soliton solutions, bright soliton solutions and periodic wave solutions corresponding to the appropriate coefficient values. We also give geometric interpretations of the obtained solutions to understand the physical meaning better. [ABSTRACT FROM AUTHOR]

In this study, the dissipation problem of nonlinear pulse in mono mode optical fibers which is governed by the Fokas system (FS) is considered. The solutions of this system have an important role in comprehending the different wave structures in physical settings. Therefore, a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the FS. The advantage of the NVTEM is to use different root possibilities of a polynomial which shape the solutions of the related model. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various exact wave solutions to the FS are obtained such as rational function, exponential function, hyperbolic function, and Jacobi elliptic function solutions. Thus, we have revealed solutions featly which are unlike the wave solutions previously found by other analytical methods. The present results depict the formation and development of such waves and their interactions. The exhibition of the solutions is given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant in achieving different wave solutions of many nonlinear differential equations in the field of optics. [ABSTRACT FROM AUTHOR]

The optical soliton solutions of the time-fractional higher-order nonlinear Schrödinger equation in the presence of the Kudryashov nonlinear refractive index are studied in this paper using the new Kudryashov approach and Bernoulli's equation approach. The current model specializes in discerning the propagation of optical soliton pulses within optical fibers. Obtaining the optical solutions for this model, particularly the sextic power, is an essential yet challenging task. To generate the nonlinear ordinary differential equation, we insert the complex wave transformations into the present time-fractional nonlinear Schrödinger equations. Here, a system of linear equations in polynomial form using the proposed approach is acquired. By solving the linear system of equations, various solution sets were generated, each containing different values for the parameters of the studied equation. Additionally, the process yielded distinct approaches for solving the problem. Several novel optical soliton solutions are constructed for the time-fractional Schrödinger equation with the Kudryashov nonlinear refractive index, and the obtained soliton solutions satisfy the model. The visual representations of the obtained solution functions through contour, three-dimensional, and two-dimensional depictions in various simulations are shown, as presented in the figures. The results suggest that the employed methods are efficient and powerful tools to be applied to various differential equations with fractional and integer orders. [ABSTRACT FROM AUTHOR]

We aim to determine the analytical solutions for the Davey–Stewartson (RDS) equations in (2+1)-dimensions using the extended sine-Gordon equation expansion method. It is a natural two-dimensional form of the nonlinear Schrödinger equation. As a first step, we change the imaginary (2+1)-dimensional RDS model to a system of nonlinear ordinary differential equations, then we apply the sine-Gordon method. With the use of Wolfram Mathematica software, we try to find novel solutions to the resulting system of the nonlinear differential equation. Constructed solutions are drawn in 2D and 3D-dimensions graphs to more realize their characteristics. [ABSTRACT FROM AUTHOR]