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In this paper, we present a novel extension of the well-known split-step Fourier transform (SSFT) approach for solving the one-dimensional nonlinear Schrödinger equation (NLSE), which incorporates the fiber loss term. While this essential equation governs the pulse propagation in a lossy optical fiber, it is not supported by an exact analytical sol...
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Citations
... For more details, see [22][23][24][25][26][27][28]. Nowadays, plenty of significant partial differential equations, such as the 2D NLSE and 2D TDSE have efficiently been solved employing either a finite difference method or a pseudo-spectral approach, such as the Gross-Pitaevskii equation [29], the complex Ginzburg Landau equation [30], 2D distributed-order timefractional cable equation [31], variable-coefficient Korteweg-De Vries equation [32], and many others [33][34][35]. In this literature, both theoretical analysis and numerical simulation are comprehensively demonstrated. ...
... This is remarkably noted due to the least achieved SSE values in terms of 10 × 10 −22 , which is almost zero, along with the smallest CPU processing time for manipulating this problem. This unique behavior might have occurred because of the employment of the splitting technique between the linear and nonlinear parts of the partial differential equation while solving them separately, in addition to the leverage of implementing the fast Fourier transform algorithm in this scheme [30,33,34]. Furthermore, it showed a perfect convergence due to exhibiting an absolute error in terms of 10 × 10 −12 . ...
In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, namely the split-step Fourier transform (SSFT), Fourier pseudo-spectral method (FPSM), and the hopscotch method (HSM). A bright 1-soliton solution is considered for the 2D NLSE, whereas a Gaussian wave solution is determined for the 2D TDSE. Although the analytical solutions of these partial differential equations can sometimes be reached, they are either limited to a specific set of initial conditions or even perplexing to find. Therefore, our suggested approximate solutions are of tremendous significance, not only for our proposed equations, but also to apply to other equations. Finally, systematic comparisons of the three suggested approaches are conducted to corroborate the accuracy and reliability of these numerical techniques. In addition, each scheme’s error and convergence analysis is numerically exhibited. Based on the MATLAB findings, the novelty of this work is that the SSFT has proven to be an invaluable tool for the presented 2D simulations from the speed, accuracy, and convergence perspectives, especially when compared to the other suggested schemes.
... The theory of optical solitons draws the attention of the researchers and scientific community, because it is an active area of research in the fields of telecommunication engineering and mathematical physics (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). The investigation of dynamic behavior of optical solitons in magneto-electroelastic (MEE) media (such as controllers, actuators, and sensors) has recently sparked a lot of interest (see [18][19][20][21]). ...
... where c 2 −r 2 m < 0. Solution-4 Substituting by Eq. (12) in (38) leads to the solution of the ODE (17) as follows: ...
... Different solutions can be obtained according to ( ) which differs according to the Eqs. (9)(10)(11)(12): ...
To get new optical soliton solutions for the nonlinear longitudinal wave equation in magneto-electro-elastic circular rod, the modified simple equation approach and the extended auxiliary equation method are implemented. New bright soliton, singular soliton and singular periodic solutions are extracted. Moreover, for more physical illustration of the extracted solutions, three- and two-dimensional plots for the solutions are presented.
In this paper, we explore applications of the third-order nonlinear Schrödinger’s equation (NLSE) using the rational exp(\(-\varphi \)(\(\zeta \)))-expansion method (REEM) and the modified exponential function method (MEFM), offering insights into wave propagation and soliton behavior in optical communications, nonlinear optics, plasma physics, quantum mechanics, and engineering. These two methods share the premise that it is first essential to use a new wave definition to transform the equation with partial derivatives into a form of the equation with ordinary derivatives. By using these methods, we are able to produce novel soliton solutions that are represented in terms of trigonometric and hyperbolic functions. The results can be utilized to comprehend and clarify the physical properties of waves propagating through a dispersive medium. Graphical shapes are generated using relevant parameter values to visually present the obtained results, including 3D, 2D, and contour plots, providing an insightful visualization of the discovered solutions. All computations in this research were exclusively conducted utilizing the symbolic software Mathematica. The discovered solutions lead to a wide range of exact solutions, including dark, bright, mixed dark-bright, periodic, singular, peakons, singular periodic soliton solutions. These solutions have diverse applications, from enhancing optical signal transmission in telecommunications to modeling complex wave behaviors in plasma physics and quantum mechanics. These fundamental, consistent, and effective methods can be used to solve a wide range of different models in physics, field of engineering, and other useful disciplines.
