Extended F-expansion Method Research Articles

Abundant families of Jacobi elliptic function solutions and peculiar dynamical behavior for a generalized derivative nonlinear Schrödinger equation employing extended F-expansion method

Abstract This study investigates a generalized derivative nonlinear Schrödinger (GDNLS) equation , demonstrating how ultrashort pulses propagate in a single-mode optical fiber. The extended F-expansion method, which is a modification of Kudryashov’s auxiliary equation approach, is applied in this investigation to generate Jacobi elliptic solutions for the GDNLS. Three distinct solution instances are examined, and a variety of explicit solutions, including breathers, solitary waves, bright/dark solitons, bright-dark interaction solitons, a soliton-like solution, and a rogue-like solution, are obtained. To demonstrate the complex dynamical behavior of GDNLS equation, several representative solutions are chosen and their moduli are shown in three-dimensional, two-dimensional, and contour plots using Maple software.

Generating optical solitons in the extended (3 + 1)-dimensional nonlinear Kudryashov’s equation using the extended F-expansion method

Generating optical solitons in the extended (3 + 1)-dimensional nonlinear Kudryashov’s equation using the extended F-expansion method

Solitons behavior of Sasa-Satsuma equation in birefringent fibers with Kerr law nonlinearity using extended F-expansion method

Optical soliton solutions in birefringent fibers and other analytic solutions for Sasa-Satsuma equation with Kerr law nonlinearity are investigated using extended F-expansion method. Bright, dark, singular, combo bright-singular, combo dark-singular and combo singular soliton solutions are extracted. Also, periodic, rational and Jacobi elliptic functions solutions are derived. Finally, we offer some 3D, 2D and contour plots to illustrate the nature of the provided solutions.

Constructing new soliton solutions for Kudryashov’s quintuple self-phase modulation with dual-form of generalized nonlocal nonlinearity using extended F-expansion method

Constructing new soliton solutions for Kudryashov’s quintuple self-phase modulation with dual-form of generalized nonlocal nonlinearity using extended F-expansion method

Exploration of New Optical Solitons in Magneto-Optical Waveguide with Coupled System of Nonlinear Biswas–Milovic Equation via Kudryashov’s Law Using Extended F-Expansion Method

Optical soliton solutions in a magneto-optical waveguide and other exact solutions are investigated for the coupled system of the nonlinear Biswas–Milovic equation with Kudryashov’s law using the extended F-expansion method. Various types of solutions are extracted, such as dark soliton solutions, singular soliton solutions, a dark–singular combo soliton, singular combo soliton solutions, Jacobi elliptic solutions, periodic solutions, combo periodic solutions, hyperbolic solutions, rational solutions, exponential solutions and Weierstrass solutions. The obtained different types of wave solutions help in obtaining nonlinear optical fibers in the future. Furthermore, some selected solutions are described graphically to demonstrate the physical nature of the obtained solutions. The results show that the current method gives effectual and direct mathematical tools for resolving the nonlinear problems in the field of nonlinear wave equations.

APPLICATION OF THE EXTENDED F-EXPANSION METHOD FOR SOLVING THE FRACTIONAL GARDNER EQUATION WITH CONFORMABLE FRACTIONAL DERIVATIVE

In this study, we derive a new fractional Gardner equation with the conformable fractional derivative for the first time. Using the extended F-expansion method, different traveling wave solutions expressed in the form of the hyperbolic solutions such as sech, tanh, csch, coth, sinh and their combination, and the triangular solutions like csc, sec, tan, cot and their combination are obtained. Finally, the solutions are illustrated through the 3D plots. The results strongly prove that the proposed approach is effective and can help us understand the nonlinear problems arising in physics better.

Construction cubic-quartic solitons in optical metamaterials for the perturbed twin-core couplers with Kudryashov's sextic power law using extended F-expansion method

In this article, we consider the twin-core couplers with Kudryashov's sextic power law of refractive index and perturbation terms. The extended F-expansion method is implemented to construct cubic–quartic solitons in optical metamaterials and other solutions for the proposed model. Bright soliton solutions, singular soliton solutions, periodic solutions and combo periodic solutions are obtained. Moreover, for the physical illustration, some of the obtained solutions are represented graphically.

Cubic-quartic solitons perturbation with couplers in optical metamaterials having triple-power law nonlinearity using extended F-expansion method

In this article, the extended F-expansion method is applied to obtain cubic-quartic optical solitons for couplers in optical metamaterials with nonlinear perturbation terms and triple-power law of nonlinearity. Bright soliton solutions, bright–singular combo soliton solutions and singular soliton solutions are extracted. Also, periodic solutions, combo periodic solutions, exponential solutions and rational solutions are derived. Moreover, for the physical illustration, some of the obtained solutions are represented graphically.

Cubic-quartic optical solitons and other solutions for twin-core couplers with polynomial law of nonlinearity using the extended F-expansion method

In this article, the extended F-expansion method is applied to secure cubic–quartic optical solitons for twin-core couplers with polynomial law of nonlinearity. Bright, singular, and bright–singular combo cubic–quartic solitons are derived. Also, periodic solutions, combo periodic solutions, exponential solutions and rational solutions are obtained. Moreover, for the physical illustration, some of the obtained solutions are represented graphically.

Bending waves of a rectangular piezoelectric laminated beam

A simple nonlinear model is proposed in this paper to study the bending wave in a rectangular piezoelectric laminated beam of infinite length. Based on the constitutive relations for transversely isotropic piezoelectric materials and isotropic elastic materials, combined with some electric conditions, we derive the bending wave equation in a long rectangular piezoelectric laminated beam by using energy method. The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal beam direction. The shock-wave solution, solitary-wave solution and other exact solutions of the bending wave equation are obtained by the extended F-expansion method. And by using the reductive perturbation method we derive the nonlinear Schrodinger (NLS) equation, further more, the bright and dark solitons are obtained. For those soliton solutions, and some parameters derived by the process of solving soliton solutions, some conclusions are drawn by numerical analysis with some fixed conditions.

Multiple Soliton Solutions for Nonlinear Differential Equations with a New Version of Extended F-Expansion Method

In this article, a new version of extended F-expansion method is suggested. A multiple Jacobi elliptic functions are presented in the solution function. We achieve analytical solutions of the (1 + 1)-dimensional dispersive long wave (DLW) equation and (2 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation using the new version of extended F-expansion method. Then, the single and the combined non-degenerative Jacobi elliptic function solutions and their degenerate solutions to the above-mentioned class of nonlinear evolution equations are obtained.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations.

Some New Nonlinear Wave Solutions for a Higher-Dimensional Shallow Water Wave Equation

In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended F-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.

Optical solitons in dual core fibers under various nonlinearities

This paper retrieves different types of soliton solutions for coupled nonlinear Schrödinger equation (NLSE). We use extended F-expansion method to find bright, dark and other trigonometric solutions with Kerr law and anti-cubic law. We also find Gausson soliton for coupled NLSE under log law nonlinearity.

Separation Transformation and a Class of Exact Solutions to the Higher-Dimensional Klein-Gordon-Zakharov Equation

The separation transformation method is extended to then+1-dimensional Klein-Gordon-Zakharov equation describing the interaction of the Langmuir wave and the ion acoustic wave in plasma. We first reduce then+1-dimensional Klein-Gordon-Zakharov equation to a set of partial differential equations and two nonlinear ordinary differential equations of the separation variables. Then the general solutions of the set of partial differential equations are given and the two nonlinear ordinary differential equations are solved by extendedF-expansion method. Finally, some new exact solutions of then+1-dimensional Klein-Gordon-Zakharov equation are proposed explicitly by combining the separation transformation with the exact solutions of the separation variables. It is shown that, for the case ofn≥2, there is an arbitrary function in every exact solution, which may reveal more nontrivial nonlinear structures in the high-dimensional Klein-Gordon-Zakharov equation.

Integrable system modelling shallow water waves: Kaup–Boussinesq shallow water system

Extended F-expansion method is proposed to seek exact solutions of the integrable nonlinear Kaup–Boussinesq shallow water system, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the extended F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously a number of periodic wave solutions expressed by various Jacobi elliptic functions for the integrable nonlinear Kaup-Boussinesq shallow water system. In the limiting cases, the solitary wave solutions are obtained as well.

Bright and dark solitons in metamaterials obtained by extended F-expansion method

The ultra-short pulse equation in a metamaterial is solved by the extended F-expansion method. The new phenomena and characteristics of solitons, caused by the anomalous self-steepening effect and the second-order nonlinear dispersion in metamaterials, are discussed. The results show that the second-order nonlinear dispersion in the positive-index region may take the place of the linear dispersion to form the bright and dark solitons. Due to the switchable sign of the anomalous self-steepening effect in the positive-index and negative-index regions, the bright and dark solitons separately exist in the anomalous and normal dispersion regions under some specific conditions. The moving directions of the centers of bright and dark solitons can be controlled by the sign of the anomalous self-steepening effect or by the combination of the anomalous self-steepening effect and third-order linear dispersion.

New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using Extended F-Expansion Method

The F-expansion method is used to find traveling wave solutions to various wave equations. By giving more solutions of the general subequation, an extended F-expansion method is introduced by Emmanuel. In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the extended F-expansion method. And when the parameters satisfy certain relations, some new exact solutions expressed by Jacobi elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

Non-TravellingWave Solutions of the (2+1)-Dimensional Dispersive Long Wave System

With the aid of symbolic computation and the extended F-expansion method, we construct more general types of exact non-travelling wave solutions of the (2+1)-dimensional dispersive long wave system. These solutions include single and combined Jacobi elliptic function solutions, rational solutions, hyperbolic function solutions, and trigonometric function solutions.

The extended F-expansion method and new exact solutions of the generalized KdV equation

In this paper, the F-expansion method is revised by adding a term with negative index with constant c，then the irregular wave solutions are avoided. By using this method， different new periodic wave solutions expressed by the generalized KdV equation are obtained.In the special cases，these periodic wave solutions degenerate to the corresponding Jacobi elliptic function solutions，solitary wave solutions or trigonometric function solutions.