G-expansion Method Research Articles

Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G′/G-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.

Optical solitons for the Hirota–Ramani equation via improved G′G-expansion method

In this work, first it is shown that the Hirota–Ramani equation, which governs the nonlinear propagation of coupled Langmuir and dust-acoustec wave in a multicomponent dusty plasma, possesses kinds of wave profile such as singular periodic profile, periodic profile, singular soliton profile, M-shape rational profile, bright soliton profile, kink profile in the form of the trigonometric, hyperbolic, and rational solutions. With the aid of symbolic computation, we select the Hirota–Ramani equation with a source to investigate the validity and advantage of the improved [Formula: see text]-expansion method and construct some frames of the 3D profiles and the contour profiles to the equation along with the 2D profiles of some solutions to understand the traveling wave dynamics. Following the selection of appropriate values for the associated parameters, more generalized solutions are provided, along with certain patterns in the solutions that are examined. This improved method is effective, concise, reliable, and can be applied for further futuristic applications.

Analytical investigations of propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide by three computational ideas

This paper mainly concentrates on obtaining solutions and other exact traveling wave solutions using the generalized G-expansion method. Some new exact solutions of the coupled nonlinear Schrödinger system using the mentioned method are extracted. This method is based on the general properties of the nonlinear model of expansion method with the support of the complete discrimination system for polynomial method and computer algebraic system (AS) such as Maple or Mathematica. The nonparaxial solitons with the propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide is studied. To attain this, an illustrative case of the coupled nonlinear Helmholtz (CNLH) system is given to illustrate the possibility and unwavering quality of the strategy utilized in this research. These solutions can be significant in the use of understanding the behavior of wave guides when studying Kerr medium, optical computing and optical beams in Kerr like nonlinear media. Physical meanings of solutions are simulated by various Figures in 2D and 3D along with density graphs. The constraint conditions of the existence of solutions are also reported in detail. Finally, the modulation instability analysis of the CNLH equation is presented in detail.

Soliton Solution of the Nonlinear Time Fractional Equations: Comprehensive Methods to Solve Physical Models

In this paper, we apply two different methods, namely, the G′G-expansion method and the G′G2-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation.

Symmetry reductions and qualitative analysis of time fractional K(m,1) equation

This paper studies symmetry reductions of time fractional K(m,1) equation with generalized evolution which is one of the important equation in fractional partial differential equations (FPDEs). Here, the classical Lie symmetries are obtained and utilized for the reduction of time fractional K(m,1) equation into ordinary differential equations. Some new exact solutions are obtained which are in the form of solitary waves by using G′G-expansion method and one of the obtained solutions is acceptable for qualitative analysis. By showing the phase portraits, its dynamical behaviour is further completely explained.

On the modified versions of G′G-expansion technique for analyzing the fractional coupled Higgs system

In this research, two modified forms of the Ḡ≡G′G-expansion method are employed to investigate various kinds of solitary wave solutions that include kink, lump, periodic, and rogue wave solutions within the framework of the fractional coupled Higgs system. The underlying patterns in the targeted model are revealed by using extended and generalized Ḡ-expansion methods. The first step involves converting the model into nonlinear ordinary differential equations via a fractional complex transformation. Following that, the suggested improved versions of the Ḡ-expansion approach are used to provide numerous solitary wave solutions. Some solitary wave solutions are represented by two- and three-dimensional graphs, demonstrating their typical propagating behavior. This research finishes by summarizing the vast findings and exploring their implications for high-energy physics.

Some Traveling Wave Solutions to the Fifth-Order Nonlinear Wave Equation Using Three Techniques: Bernoulli Sub-ODE, Modified Auxiliary Equation, and G ′ / G -Expansion Methods

The fifth-order nonlinear wave equation contains terms involving higher-order spatial derivatives, such as u x x x and u x x x x x . These terms are responsible for dispersion, which affects the shape and propagation of the wave. The study of dispersion is important in many areas, including seismology, acoustics, and communication theory. In the current work, three potent analytical techniques are proposed in order to solve the fifth-order nonlinear wave equation. The used approaches are the modified auxiliary equation method, the Bernoulli Sub-ODE method, and the G ′ / G -expansion method (MAE). Some graphs are plotted to display our findings. The solutions to the nonlinear wave equation are used to describe the nonlinear dynamics of waves in physical systems. The results show how the dynamics of the wave solutions are influenced by the system parameters, which can be used as system controllers. The new approaches used in this work helped to find new solutions for traveling waves. This could be seen as a new contribution to the field. Water waves, plasma waves, and acoustic wave behavior can be described by the obtained solutions.

Exact Solutions of an Extended Jimbo-Miwa Equation by Three Distinct Methods

In this article, we focus on exact traveling wave solutions to an extended Jimbo-Miwa equation, which is an extension of the Jimbo-Miwa equation. First, an improved G ′ / G -expansion method, extended G ′ / G -expansion method, and improved two variable ( φ ′ / φ , 1 / φ ) expansion method are introduced. Second, with these introduced methods, many new exact traveling wave solutions of EJM equation are constructed, including hyperbolic function solutions, trigonometric function solutions, and rational function solutions which contain many different parameters. Finally, we depict the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. To the best of our knowledge, the received results have not been reported in other studies on the new extended JM equations. We hope that our results can help enrich the study of this new equation.

Analytical and numerical solutions with bifurcation analysis for the nonlinear evolution equation in (2+1)-dimensions

This paper is focused on the nonlinear evolution equation in (2+1)-dimensions which is found in different engineering and scientific areas. Many sets of exact soliton solutions of the nonlinear evolution equation in (2+1)-dimensions are presented via two analytical methods such as the Kudryashov method and the G′G-expansion method. Numerical solutions based on the finite difference method are introduced. The bifurcation analysis of solitary wave solutions is exhibited. Our analytical and numerical results are compared and illustrate them through some tables and graphical figures.

On nonlinear optical solitons of fractional Biswas-Arshed Model with beta derivative

The Biswas-Arshed (BA) model is an essential wave propagation model in nonlinear optics and optical fibers. In this manuscript, the enhanced G′/G-expansion method is taken into consideration. The BA model is used with this technique to locate visual soliton solutions. The solutions to the hyperbolic and trigonometric functions are determined using this technique. Each solution includes a large number of model and method parameters. The wave solutions are defined by the waves' character, which is determined by the three-dimensional (3D) and two-dimensional (2D) wave patterns with varying parameter values. The structure of the soliton shifts depending on the specific values of the parameters. These shifts are reflected in the 2D pictures that demonstrate the various soliton configurations. Finally, the obtained soliton solutions can be applied to high-speed data transmission to describe propagation pulses and to transmit light between two ends in optical fibers and communication systems.

Exact Solutions of the Generalized ZK and Gardner Equations by Extended GeneralizedG′/G-Expansion Method

In this investigation, the exact solutions of variable coefficients of generalized Zakharov-Kuznetsov (ZK) equation and the Gardner equation are studied with the help of an extended generalizedG′/Gexpansion method. The main objective of this study is to establish the closed-form solutions and dynamics of analytical solutions to the generalized ZK equation and the Gardner equation. The generalized ZK equation and the Gardner equation govern the behavior of nonlinear wave phenomena in the presence of magnetic field in plasma dynamics, turbulence, bottom topography, and quantum field theory. We construct innovative solutions to the models under consideration using various computing tools and a recently developed extended generalizedG′/Gexpansion technique. The extended generalizedG′/Gexpansion technique is a well-defined and simple technique which is based on the initial assumed solutions of the polynomial ofG′/G. The derived solutions for both the equations are the hyperbolic, trigonometric, and rational functions. The obtained solutions have shock/kink waves and multisoliton, which depict the dynamical representations of the acquired solutions through the three-dimensional surface plots and the contour plots.

Bright and dark solitons for the fifth-order nonlinear Schrödinger equation with variable coefficients

In this work, a fifth-order nonlinear Schrödinger equation with variable coefficients is investigated, which represents the diffusion of ultrashort pulses in incompatible optical fibers. With the aid of two direct functions, some non-traveling wave bright and dark soliton solutions are obtained. Via the G′/G-expansion method, the non-traveling wave hyperbolic-type and trigonometric-type solutions are studied. Finally, we show the dynamic properties and characteristics of these derived results by using some three-dimensional figures and contour figures.

Soliton solutions to the electric signals in telegraph lines on the basis of the tunnel diode

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.

Exact solutions of a quintic dispersive equation

We present novel exact solutions for a class of Rosenau’s quintic dispersive equations. The variational derivative approach is employed to construct conservation laws for a slightly generalized version of the quintic equation. The double reduction theory, based on the association of symmetries and conservation laws, is utilized to obtain a fourth-order nonlinear ODE, which is then solved to compute exact solutions of the quintic equation. In particular, the G′G-expansion method for the reduced fourth-order nonlinear ODE is applied to construct new exact solutions of the quintic PDE with a cubic nonlinearity.

Applications of Two Methods in Exact Wave Solutions in the Space-Time Fractional Drinfeld–Sokolov–Wilson System

The fractional differential equations (FDEs) are ubiquitous in mathematically oriented scientific fields, such as physics and engineering. Therefore, FDEs have been the focus of many studies due to their frequent appearance in several applications such as physics, engineering, signal processing, systems identification, sound, heat, diffusion, electrostatics and fluid mechanics, and other sciences. The perusal of these nonlinear physical models through wave solutions analysis, corresponding to their FDEs, has a dynamic role in applied sciences. In this paper, the exp-function method and the rational G ′ / G -expansion method are presented to establish the exact wave solutions of the space-time fractional Drinfeld–Sokolov–Wilson system in the sense of the conformable fractional derivative. The fractional Drinfeld–Sokolov–Wilson system contains fractional derivatives of the unknown function in terms of all independent variables. This system describes the shallow water wave models in fluid mechanics. These presented methods are a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences, especially in physics.

Soliton solutions and fractional effects to the time-fractional modified equal width equation

The one-dimensional long water wave propagation in a nonlinear medium, including the dispersion process, is well simulated by the fractional-order modified equal-width (MEW) equation. This article establishes several recognized, standard, inclusive, and scores of typical exact wave solutions to the MEW equation using the double G'/G,1/G-expansion method. For specific parameter values, kink, periodic, periodic-singular, singular-kink, and other forms of solitons can be recovered from general solutions. The effect of the fractional parameter on wave forms has also been analyzed by depicting several graphs for different values of the fractional-order α. In order to illustrate the potential characteristics, three- and two-dimensional combined plots using Maple have been drawn. It has been established that the introduced approach is a potential tool for extracting new exact solutions to various nonlinear evolution equations (NLEEs) arising in engineering, science, and applied mathematics.

Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation

In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the Riemann–Liouville fractional derivative is derived by the semi-inverse method and variational method. Further, the symmetry of the time-fractional equation is obtained by the fractional Lie symmetry analysis method. Based on the symmetry, the conservation laws of the time fractional equation are constructed by the new conservation theorem. Then, the G′G-expansion method is used here to solve the equation and obtain the exact traveling wave solutions. Finally, the spectral method in the spatial direction and the Gru¨nwald–Letnikov method in the time direction are considered to obtain the numerical solutions of the time-fractional equation. The numerical solutions are compared with the exact solutions, and the error results confirm the effectiveness of the proposed numerical method.

Some new type optical and the other soliton solutions of coupled nonlinear Hirota equation

In this study, we obtained some exact traveling wave solutions of coupled nonlinear Hirota equation (CNHE) by using generalized G'G-expansion method. The solutions we obtained are hyperbolic, trigonometric and exponential solutions. Analytical results for the obtained solutions for the CNHE have been corroborated through the Mathematica 11.2 symbolic calculation. Solutions have been illustrated in 3D graphs, and 2D plots. The present method has been used to obtain exact traveling wave solutions for several nonlinear partial differential equations.

On the Exact Solutions of Nonlinear Potential Yu–Toda–Sasa–Fukuyama Equation by Different Methods

In this article, the exact solutions to the potential Yu–Toda–Sasa–Fukuyama equation are successfully examined by the extended complex method and (G′/G)‐expansion method. Consequently, we find solutions for three models of Weierstrass elliptic functions, simply periodic functions, and rational function solutions. The obtained results will play an important role in understanding and studying potential Yu–Toda–Sasa–Fukuyama equation. It is observed that the extended complex method and (G′/G)‐expansion method are reliable and will be used extensively to seek for exact solutions of any other nonlinear partial differential equations (NPDEs).

White Noise Functional Solutions for Wick-Type Stochastic Fractional Mixed KdV-mKdV Equation Using Extended G ′ / G -Expansion Method

In this paper, white noise functional solutions of Wick-type stochastic fractional mixed KdV-mKdV equations have been obtained by using the extended G ′ / G -expansion method and the Hermite transform. Firstly, the Hermite transform is used to transform Wick-type stochastic fractional mixed KdV-mKdV equations into deterministic fractional mixed KdV-mKdV equations. Secondly, the exact traveling wave solutions of deterministic fractional mixed KdV-mKdV equations are constructed by applying the extended G ′ / G -expansion method. Finally, a series of white noise functional solutions are obtained by the inverse Hermite transform.