Extended Tanh-function Method Research Articles

Soliton dynamics and travelling wave solutions for high-order nonlinear Schrödinger equation in birefringent fibers using improved modified extended tanh function method

Soliton dynamics and travelling wave solutions for high-order nonlinear Schrödinger equation in birefringent fibers using improved modified extended tanh function method

Unraveling solitons dynamics in system of dispersive NLSE with Kudryashov's law of nonlinearity using improved modified extended tanh function method

In this work, the dynamics for coupled system of dispersive nonlinear Schrödinger equation with Kudryashov's law of nonlinearity are studied with the aid of improved modified extended tanh function method. This model is used to mimic the wave propagation through magneto-optic wave guides. Various types of solutions are extracted for the proposed model. These solutions including dark solitons, bright solitons and singular solitons, Weierstrass elliptic and singular periodic solutions. Moreover, the graphical illustrations of some solutions are introduced to demonstrate the nature of them.

Optical solitons and other wave solutions to fractional-order nonlinear Sasa–Satsuma equation in mono-mode optical fibers using IM extended tanh function method

Optical solitons and other wave solutions to fractional-order nonlinear Sasa–Satsuma equation in mono-mode optical fibers using IM extended tanh function method

Analytical soliton solutions for cubic-quartic perturbations of the Lakshmanan-Porsezian-Daniel equation using the modified extended tanh function method

In this paper, the improved modified extended Tanh function technique is used to find analytical solutions to the cubic-quartic Lakshmanan-Porsezian-Daniel model which describes the pulse propagation in optical fiber. Using the traveling wave theory, the nonlinear partial differential form of the cubic-quartic Lakshmanan-Porsezian-Daniel equation is transformed into an ordinary differential equation. Then applying the proposed methods bright, dark, bright-dark, periodic, singular, and Jacobian elliptic solutions were derived as analytical solutions. Besides, hyperbolic, rational, trigonometric, and exponential functions were also obtained. In order to understand the obtained solutions, graphical simulations were depicted by 3D surfaces and 2D graphs.

Novel soliton solutions and localized structures of KdV equation

A (1 + 1)-dimensional Korteweg-de Vries (KdV) equation is studied. Using the extended Tanh-function method, the analytical soliton solutions are obtained by symbolic calculation, and the localized coherent structures containing the compacton, the peakon, periodic multi-solitons and fractal soliton structures are constructed. Through the analysis of various images, the properties of soliton solutions excited by different local structures are discussed and the fractal structure of soliton solutions are further discussed.

Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative

In this work, we consider the Boiti–Leon–Manna–Pempinelli equation with the M-truncated derivative (BLMPE-MTD). Our aim here is to obtain trigonometric, rational and hyperbolic solutions of BLMPE-MTD by employing two diverse methods, namely, He’s semi-inverse method and the extended tanh function method. In addition, we generalize some previous results. As the Boiti–Leon–Manna–Pempinelli equation is a model for an incompressible fluid, the solutions obtained may be utilized to represent a wide variety of fascinating physical phenomena. We construct a large number of 2D and 3D figures to demonstrate the impact of the M-truncated derivative on the exact solution of the BLMPE-MTD.

Novel analytical solutions of stochastic Ginzburg-Landau equation driven by Wiener process via the improved modified extended tanh function method

In this manuscript, the improved modified extended tanh integration technique is implemented to investigate the exact solutions of stochastic Ginzburg–Landau model driven by Wiener process which appeared in various fields of chemistry, mathematics and physics. Many types of stochastic wave solutions are raised such as hyperbolic stochastic solutions, trigonometric stochastic solutions, Weierstrass elliptic stochastic solutions and Jacobi elliptic stochastic solutions. Moreover, 3D and 2D graphical representations are depicted to show the effect of the multiplicative noise on the propagated waves. The results reveal that as the noise level increases, the wave degrades. The wave is distorted when σ⩾1. In addition, the 3D graphical representation show that the propagated wave is flattered when the noise strength is increased. These solutions may be applicable in various applications in applied science. The proposed method is efficient, direct and powerful.

Solitary Wave Solutions to a Fractional Model Using the Improved Modified Extended Tanh-Function Method

Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in understanding the long-term dynamics of these events. This article primarily focuses on using the improved modified extended tanh-function algorithm to determine certain traveling wave solutions to the space-time fractional symmetric regularized long wave (SRLW) equation, which is used to discuss space-charge waves, shallow water waves, etc. The Jumarie’s modified Riemann-Liouville derivative is successfully used to deal with the fractional derivatives, which appear in the SRLW problem. We find many traveling wave solutions on the form of trigonometric, hyperbolic, complex, and rational functions. Furthermore, the performance of the employed technique is investigated in comparison to other techniques such as the Oncoming exp(−Θ(q))-expansion method and the extended Jacobi elliptic function expansion strategy. Some obtained results are graphically displayed to show their physical features. The findings of this article demonstrate that the used approach enables us to handle more NLFPDEs that emerge in mathematical physics.

Exploration of new solitons in optical medium with higher-order dispersive and nonlinear effects via improved modified extended tanh function method

Throughout this article, the improved modified extended tanh function method is presented for deriving fresh travelling wave alternative solutions in different forms of nonlinear Schrödinger equation (NLSE) with higher-order dispersive and nonlinear effects that describes the propagation of ultra-short optical solitons in an optical fiber medium. For this model, we have a variety of new solutions, including solitons (bright, dark and singular), exponential, singular periodic and Jacobi elliptic solutions. Graphical representations of selected solutions are provided to demonstrate the physical properties of the raised solutions.

Soliton Waves with the (3+1)-Dimensional Kadomtsev–Petviashvili–Boussinesq Equation in Water Wave Dynamics

We examined the (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which arises not only in fluid dynamics, superfluids, physics, and plasma physics but also in the construction of connections between the hydrodynamic and optical model fields. Moreover, unlike the Kadomtsev–Petviashvili equation (KPE), the KP-B equation allows the modeling of waves traveling in both directions and does not require the zero-mass assumption, which is necessary for many scientific applications. Considering these properties enables researchers to obtain more precise results in many physics and engineering applications, especially in research on the dynamics of water waves. We used the modified extended tanh function method (METFM) and Kudryashov’s method, which are easily applicable, do not require further mathematical manipulations, and give effective results to investigate the physical properties of the KP-B equation and its soliton solutions. As the output of the work, we obtained some new singular soliton solutions to the governed equation and simulated them with 3D and 2D graphs for the reader to understand clearly. These results and graphs describe the single and singular soliton properties of the (3+1)-dimensional KP-B equation that have not been studied and presented in the literature before, and the methods can also help in obtaining the solution to the evolution equations and understanding wave propagation in water wave dynamics.

Soliton Solutions and Other Solutions for Kundu–Eckhaus Equation with Quintic Nonlinearity and Raman Effect Using the Improved Modified Extended Tanh-Function Method

Our paper studies the optical solitons for the Kundu–Eckhaus (KE) equation with quintic nonlinearity and Raman effect. By applying the improved modified extended tanh-function method, many soliton solutions can be obtained such as bright soliton solutions, dark soliton solutions, and the singular soliton solution. In addition, we can obtain various types of solutions, namely, singular periodic solutions, exponential solutions, rational solutions, Jacobi elliptic solutions and Weierstrass elliptic doubly periodic solutions. Moreover, some selected solutions are illustrated graphically to show the physical nature and the characteristics of the obtained solutions.

Construction Cubic–Quartic Optical Solitons and Other Solutions for Coupled System of Fokas–Lenells Equation Using the Improved Modified Extended tanh Function Method

Construction Cubic–Quartic Optical Solitons and Other Solutions for Coupled System of Fokas–Lenells Equation Using the Improved Modified Extended tanh Function Method

Construction of solitons and other solutions for NLSE with Kudryashov’s generalized nonlinear refractive index

In our study we implement an improved modified extended tanh-function (IMETF) method for constructing new solitons and other types of solutions with NLSE having Kudryashov’s generalized nonlinear refractive index. Bright, dark and singular optical soliton solutions have been recovered for the model by this approach. Moreover, 3D and contour graphs are plotted for more illustration.

Extended Tanh-Function Method and Its Applications in Nonlocal Complex mKdV Equations

In order to construct the multiple traveling wave solutions of the nonlocal modified Korteweg de Vires (mKdV) equation, the modified tanh-function approach for local soliton equations is extended to a nonlocal complex mKdV equation. The central idea of this method is to use the solution of the Riccati equation to replace the tanh function in the tanh function (THF) method. As an application, we investigate a new traveling wave solution for the nonlocal complex mKdV equation of Ablowitz and Musslimani. Moreover, some exciting diagrams show the underlying dynamics of some given solutions.

The improved modified extended tanh-function method to develop the exact travelling wave solutions of a family of 3D fractional WBBM equations

The improved modified extended tanh-function (imETF) method is used to develop exact traveling wave solutions of a family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations. Bright solitons, dark solitons, bright-dark solitons, single solitons, and multiple solitons are obtained by using the imETF method. Also, periodic solutions, singular periodic solutions, hyperbolic function solutions, rational function solutions, trigonometric function solutions, and exponential function solutions are reported. Through companionable wave transformation, the governing equations are simplified to conventional ordinary differential equations. The alternative solution relates to the earlier phase. The anticipated solution's power coefficients are compared to create a system of algebraic equations (SAE). The links between parameters and coefficients that are required to create solutions are provided by a resilient coefficients scheme. For various parameter combinations, certain solutions are modeled.

Perturbation of dispersive optical solitons with Schrödinger–Hirota equation with Kerr law and spatio-temporal dispersion

Objective:The principal purpose of this paper is to examine the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity which governs the propagation of dispersive pulses in optical fibers by proposing and using a direct algebraic form of the enhanced modified extended tanh expansion method for the first time. Our aim is not only restricted to obtaining different and more soliton solutions by proposed method for the first time in this study, but also includes examining the effect of the coefficients of self-steepening and nonlinear dispersion terms to the soliton propagation in the investigated problem. Methodology:Utilizing a traveling wave transformation, the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity can be transformed into an nonlinear ordinary differential equation (NODE). Then, the NODE is convert into a set of algebraic equations by taking account into the Riccati differential equation. Solving the set of algebraic equations, we acquire the analytical soliton solutions of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. In the proposed method, the modified extended tanh function method is enhanced by presenting more solutions of Riccati differential equations with the direct algebraic form, is utilized. Results:The more solutions have been established to the literature with new significant physical properties of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. We indicated that the presented method are effective, easily computable, and reliable in solving such nonlinear problems. Moreover, we demonstrate the dynamical behaviors and physical significance of some soliton solutions at appropriate values of parameters. Originality:A variety of soliton solutions to the perturbed Schrödinger–Hirota equation with Kerr law non-linearity by the direct algebraic form of enhanced modified extended tanh expansion method have been acquired. These solutions are dark–bright, trigonometric, hyperbolic, periodic, and singular soliton solutions. 3D, contour and 2D plots of some obtained solutions have been demonstrated to interpret the physical meaning of the equation. For some parameter values in the equation, the behavior of soliton solutions has been examined. The constraint conditions are established to confirm the existence of valid solutions. The obtained results can be effective in interpreting the physical meaning of this nonlinear system. We have seen that the proposed direct algebraic form of the enhanced modified extended tanh expansion method is a powerful mathematical technique which can be utilized to acquire the analytical solutions to different complex nonlinear mathematical models.

Constructing new solitary wave solutions to the strain wave model in micro-structured solids

In the present article, the improved modified extended tanh function method is implemented to investigate the strain wave model which governs the wave propagation in micro-structured solids. Solitary and other wave solutions are provided for this model such as singular and bright solitary type solutions. In addition Kink, periodic, Jacobi elliptic, hyperbolic, exponential and Weierstrass elliptic type solutions are constructed. To show the physical characteristics of the raised solutions, graphical illustration of some solutions is presented.

Computational extracting solutions for the perturbed Gerdjikov-Ivanov equation by using improved modified extended analytical approach

In this work, the improved modified extended tanh function method will be applied to study the perturbed Gerdjikov-Ivanov equation (PGI) which describe the dynamics of the soliton in optical fibers. New exact solutions for the PGI equation will be introduced including Weierstrass elliptic, Jacobi elliptic, dark and bright solitons, singular periodic and rational type solutions. Moreover, for the physical illustration of the obtained solutions, 3D and 2D graphs are presented.

Optical solitons and other solutions for coupled system of nonlinear Schrödinger’s equation with parabolic nonlocal law of refractive index by using the improved modified extended tanh function method

In this paper, the improved modified extended tanh function method is applied to secure optical soliton solutions in magneto-optic waveguides for coupled system of non-linear Schrödinger equation with parabolic non-local law of refractive index. Bright soliton solutions, dark soliton solutions and singular soliton solutions are obtained. Moreover, for the physical illustration of the obtained solutions, 3D and their corresponding 2D graph are presented.

The Improved Modified Extended Tanh-Function Method to Develop the Exact Travelling Wave Solutions of a Family of 3d Fractional Wbbm Equations

The Improved Modified Extended Tanh-Function Method to Develop the Exact Travelling Wave Solutions of a Family of 3d Fractional Wbbm Equations