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In this study, the dispersal caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod is taken into consideration using the nonlinear longitudinal wave equation (LWE), a mathematical physics problem. Using the generalized exp-function method, we investigate the families of solitary wave solutions of one-dimensional...
Citations
... As a result, we secured new, more general, interested set of optical soliton solutions. If we compare the differences and similarities of our work with those already set up in the past literatures we see that many researchers have resolved nonlinear longitudinal wave equation utilized the different techniques like generalized algebraic approach , extended trial equation approach (Seadawy and Manafian 2018), modified exp (− ( )) function approach (Baskonus et al. 2016), mathematical methods , (1∕G � ) expansion approach (Durur et al. 2021), the Jacobi elliptic function expansion approach (Xue et al. 2011), modified Sardar subequation approach (Younas et al. 2022), the sine Gordan equation approach (Ilhan et al. 2017), generalized Exp-function approach (Shakeel et al. 2022), improved Bernouli sub equation function approach (Baskonus and Gómez-Aguilar 2019), auxiliary equation mapping method (Sajid and Akram 2021) are applied to this equation and obtained results in hyperbolic, complex, hyperbolic, trigonometric, periodic, rational, elliptic functions and also in dark solitons, bright solitons, solitary waves, travelling waves, and periodic solitary waves. But our results represent to dark solitons, kink wave solitons, bright solitons, singular solitons, combined bright-dark solitons, periodic singular solitons, anti-kink wave solitons, and solitary wave solutions. ...
In this research work, we utilized the auxiliary equation technique and extracted the optical soliton solutions to the nonlinear longitudinal wave equation (NLLWE) in magneto electro elastic (MEE) rod that was spread out in a circle. The NLLWE in MEE systems deals with the mathematical physics of transverse Poisson’s effect dispersal and also very important in many engineering fields like sensors and actuators. As a result, we extracted the exact soliton solutions in bright solitons, dark solitons, kink wave solitons, anti-kink wave solitons, combined bright-dark solitons, solitary waves and periodic singular solitons. The physical structure of some extracted solutions visualizing in contour, two, and three dimensions through numerical simulation. The explored soliton solutions are interested, more general and having different physical structure, which may will be helpful to study of physical phenomena in the fields of optical fibers, plasma physics, soliton wave theory, nonlinear optics, ocean engineering, nonlinear dynamics and different branches of applied sciences. The successful extraction of exact solitons shows that this utilized approach is effective, straightforward, concise and powerful can also applicable to other nonlinear partial differential equations that involve in mathematical physics, engineering and applied sciences.
... The recent developments in the field of soliton theory and its applications are studied in Shakeel et al. (2023a), Zhang et al. (2023), Shakeel et al. (2023b), , Shakeel et al. (2022) and Attaullah et al. (2022). A variety of explicit methods are employed in solving important and practical problems (Ghanbari and Gómez-Aguilar 2019a, b;Khater and Ghanbari 2021;Ghanbari 2019;Ghanbariand and Akgül 2020;Kuo and Ghanbari 2019;Ghanbari and Baleanu 2019) in recent years. ...
In this research paper, the main aim is to investigate the nonlinear optical transmission equation (NOTE) with two explicit methods. The NOTE is a nonlinear partial differential equation that describes the propagation of light waves in a nonlinear medium, such as an optical fiber or a waveguide. The equation takes into account the effects of dispersion, diffraction, nonlinearity, dissipation, and external potential on the evolution of the complex envelope field of the light wave. The extended (G′G2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Big (\frac{G'}{G^{2}}\Big )$$\end{document}-expansion method and exp(-ϕ(η))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp (-\phi (\eta ))$$\end{document} expansion method are successfully applied on the proposed model. The new and novel soliton solutions and other solutions are obtained by the applications of these methods. These solutions include dark soliton, singular soliton, periodic solutions and rational solutions. Constraint conditions for the existence of obtained solutions are also provided in this article. Graphically, the retrieved solutions present different shapes of wave propagations which is shown by 3D graphs. These methods are being applied for the first time on the proposed model and considered as most recent techniques to obtain the soliton solutions for the proposed model.
... In recent decades, with the rapid development of nonlinear science, the soliton theory, as an important branch, has also developed rapidly. Soliton theory has been widely applied in various fields such as physics, photochemistry, and quantum mechanics [1][2][3][4][5][6]. During the exploration and solution of nonlinear partial differential equations (NPDEs), many methods have been developed for solving nonlinear evolution equations (NLEEs). ...
... To better understand the physical properties of the bilinear equation, we construct a 4-4-1 model to explore the interactions between lump solutions and double solitons. We set the activation function 4 , and the constructed neural network model is shown in Fig. 3a, b. The constructed structure is ...
The (3+1)-dimensional BKP-Boussinesq equation is widely used to describe and understand nonlinear wave phenomena. In this article, a single hidden layer neural network model is constructed using the bilinear neural network method to obtain lump solutions, interaction solutions, and breather solutions of the equation. Based on the single-layer model, a ’4-2-2-1’ neural network model was constructed. By assigning different weight coefficients and thresholds, interference wave solutions and periodic solutions of the equations are obtained. During the solving process, some weight coefficients being zero may lead to the degeneration of the bilinear neural network model, and this phenomenon can be mitigated by appropriately enhancing the model’s performance. Furthermore, the study shows that due to the universal approximation property of neural networks, the bilinear neural network method offers a more flexible and simpler way to solve nonlinear problems. It can yield a greater number of novel analytical solutions and promote the development of the generality of solving nonlinear partial differential equations.
... Recent interest has been growing in investigating soliton equations and their hierarchies with self-consistent sources. These sources appear in solitary waves with alternating speeds, leading to diverse dynamics in physical models such as plasma physics, hydrodynamics, and solid-state physics [14][15][16][17][18][19][20][21][22][23][24][25][26]. The work [17] is dedicated to Korteweg- ...
This work presents an algorithm that uses the inverse scattering method to find a solution for the higher-order Toda lattice with a self-consistent source. The higher-order Toda lattice with an integral-type source is also a significant theoretical model belonging to very integrable systems. The problem is solved by applying the direct and inverse scattering methods to the discrete Sturm-Liouville operator, and the time dependence of the scattering data for this operator is attained. The solution to the problem is set up using the inverse scattering transform (IST) approach.
... The current technique could be implemented in other fields of science and engineering, especially when related to the simulation of fluid dynamics in the future [28][29][30][31][32][33][34]. ...
Citation: Brahim, M.; Benhanifia, K.; Jamshed, W.; Al-Farhany, K.; Redouane, F.; Eid, M.R.; Hussain, S.M.; Akram, M.; Kamel, A.
... During the past serval decades, there have been many analyzes on the fifth-order NLS equation, in which soliton and periodic solutions have been extensively studied. Soliton solutions are caused by the cancellation of nonlinear and dispersion effects in media, usually propagates at a constant rate and maintains its shape, and can describe the solution of a class of weakly nonlinear dispersion partial differential equations widely existing in physical systems [19][20][21][22][23]. Equation (1.1) has been proved to be integrable when the coefficients satisfy certain constraints, the Lax pair and other required complete integrable properties have been recognized, and the exact expressions of soliton solutions have been obtained by the Darboux transformation [24]. ...
In this paper, we investigate the periodic solutions and Whitham modulation theory for the fifth-order nonlinear Schrödinger equation, which can describe the one-dimensional anisotropic Heisenberg ferromagnetic spin chain. First, we introduce the principle of the finite-gap integration method. Then, we discuss the single-phase periodic solutions and their degenerate forms in two limit cases. In addition, we analyze the influence of higher-order term parameters on the propagation of periodic solutions and solitons. Further, we derive the single-phase Whitham modulation equation for the fifth-order nonlinear Schrödinger equation. Moreover, we systematically derive the two-phase periodic solutions and the corresponding Whitham modulation equations.
... The SSs gotten as an effect of implementing these methods permit observing on the performance of NLEMs. Some of them are M ′ ∕ EM, 1-5 the exponential function technique, 6 variational derivative method, 7 the improved simple equation scheme, 8 modified advanced direct algebraic process, 9 the Jacobi elliptic task extension scheme, 10 Sine-Gordon expansion way, 11 Cubic B-spline scheme, 12 the unified method, 13,14 the (1/G ′ )-expansion scheme, 15 sub-equation scheme, 16 simplified Hirota's method, 17 novel computational technique, 18 the Lie symmetry investigation, 19 24 the exp(phi(zta))-expansion method, 25 modified exp-function method, [26][27][28][29][30] new extended direct algebraic method and, 31 Hirota bilinear approach, 32,33 the extended sinh-Gordon equation expansion method, 34 complex envelop ansatz, 35 Adomian decomposition method, 36 sine-Gordon expansion method, 37 the modified Sardar sub-equation method, 38 rational ansatz method, 39 Hirota direct method, [40][41][42][43] Lie Symmetry analysis method, 44 the modified Khater method, 45 hybrid B-spline collocation technique 46 and so many. ...
Soliton solutions (SSs) of the electric signals in telegraph lines on the basis of the tunnel diode (ESTLBTD) have been examined using the modified G′/G-expansion method (MG′/GEM). We create general SSs that define rational, hyperbolic and trigonometric solutions. The 3D, 2D and contour plots of the gained SSs are represented to explain the ESTLBTD in the optical fibers. Equating our attained answers and that gotten in beforehand written examination papers offerings the novelty of our study. The gained SSs might show imperative part in nonlinear science and engineering (NLSE) areas. It is remarkable to notice that the MG′/GEM is easy, well-matched and influential scientific device to determine SSs for nonlinear engineering models (NLEMs). The study procedures could also be implemented to develop SSs for other NLEMs in mathematical physics, applied mathematics, and engineering.
... The current technique could be implemented in other fields of science and engineering, especially when related to the simulation of fluid dynamics in the future [28][29][30][31][32][33][34]. ...
Several industrial fields require mixing and mechanical agitation processes. This operation is mainly used to enhance heat and mass transfer inside stirred tank systems and improve the degree of homogeneity to obtain a high-quality final product. The main goal of this research paper is to analyze the thermal and hydrodynamic behavior of non-Newtonian nanofluid (Bingham– Papanastasiou–Al2O3) inside a symmetrically stirred tank. A 3D numerical study has been conducted for a stationary laminar flow inside a symmetric cylindrical vessel under influencing parameters, including the inertia parameter (Re = 1, 20, 100) and the volume fraction of nanoparticles
(Ø = 0.02, 0.06, 0.1) with different geometric configurations, has been introduced into the stirring system. According to the findings, with high inertia (Re = 100), the heat transfer inside the stirred tank is enhanced. Furthermore, increasing the nanoparticle fraction volume had a significant impact on the acceleration of heat transfer along the stirred vessel. It has been also found that the geometric configuration of an anchor with added arm blade (Case 2) is more efficient compared with the rest of the anchor agitator.
This study purposes to extract some fractional analytical solutions of the Peyrard-Bishop-Deoxyribo-Nucleic-Acid ([Formula: see text]-PBDNA) dynamic model with the beta-derivative by the unified Riccati equation expansion method (UREEM). Furthermore, we examine the role of various parameters of the fractional model on the soliton dynamic. The research focuses on computational biophysics and materials science, examining the impact of various parameters on the fractional model. This paper contributes to understanding soliton solutions and the [Formula: see text]-PBDNA dynamic model, demonstrating the applicability of the UREEM method to various fractional models. Some soliton solutions of the model are successfully generated by applying the UREEM. Implementing the UREEM, we take a fractional wave transformation to convert the model into a nonlinear ordinary differential equation. So, a linear equation system is generated. After the system is solved, the soliton solutions are gained for the appropriate solution sets. Finally, 3D, 2D and contour graphs of diverse solutions are depicted at suitable values of parameters. In addition, this paper presents 3D, 2D and contour graphs of various solutions with suitable parameter values. The results are beneficial for interpreting the model in future work and confirm that UREEM is effectively applicable to diverse fractional models, coupled with a comprehensive graphical analysis of how different parameters influence these solutions.
This study focuses on analyzing a coupled space-time fractional nonlinear Schrodinger equation, which has applications in describing non-relativistic quantum mechanical behavior. The investigation covers various aspects, including the examination of dynamical behaviors and the exploration of optical soliton solutions. The modified F-expansion method is employed to derive these soliton solutions. To visualize and interpret the physical characteristics of the solutions, they are plotted in 2D, 3D, and density plots with appropriate parameter settings. The dynamical behaviors of the equation are discussed by investigating bifurcations at equilibrium points, and the chaotic behavior of the perturbed dynamical system is demonstrated using chaos theory. Phase portraits illustrating bifurcation and chaotic patterns are generated using the RK4 algorithm in Matlab. These findings provide a dynamic and powerful mathematical tool to address a range of nonlinear wave phenomena. Key discoveries include the identification of new solitary wave forms, as well as bifurcation and chaotic solutions. These unique and intriguing solutions have theoretical significance in understanding energy transfer and diffusion processes in mathematical models.
