Exponential Rational Function Method Research Articles

Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

Exact solitary wave propagations for the stochastic Burgers’ equation under the influence of white noise and its comparison with computational scheme

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

A diverse array of optical solitons in the damped (2 + 1)-dimensional nonlinear Schrödinger equation via the modified exponential rational function method and other distinct strategies

A diverse array of optical solitons in the damped (2 + 1)-dimensional nonlinear Schrödinger equation via the modified exponential rational function method and other distinct strategies

Extraction of new soliton solutions of (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation via generalized exponential rational function method and $$\left( \frac{G^{\prime }}{G},\frac{1}{G}\right) $$ expansion method

Extraction of new soliton solutions of (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation via generalized exponential rational function method and $$\left( \frac{G^{\prime }}{G},\frac{1}{G}\right) $$ expansion method

Generalized exponential rational function method for solving nonlinear conformable time-fractional Hybrid-Lattice equation

Generalized exponential rational function method for solving nonlinear conformable time-fractional Hybrid-Lattice equation

Analyzing Soliton Solutions of the Extended (3 + 1)-Dimensional Sakovich Equation

This work focuses on the utilization of the generalized exponential rational function method (GERFM) to analyze wave propagation of the extended (3 + 1)-dimensional Sakovich equation. The demonstrated effectiveness and robustness of the employed method underscore its relevance to a wider spectrum of nonlinear partial differential equations (NPDEs) in physical phenomena. An examination of the physical characteristics of the generated solutions has been conducted through two- and three-dimensional graphical representations.

Dynamics of M-truncated optical solitons and other solutions to the fractional Kudryashov’s equation

The study of soliton theory plays a crucial role in the telecommunication industry’s utilization of nonlinear optics. The principal area of research in the field of optical solitons revolves around optical fibers, metamaterials, metasurfaces, magneto-optic waveguides, and other related technologies. Therefore, the examination of these solitons received significant attention from scholars in recent years. Optical solitons refer to electromagnetic waves that are confined within nonlinear dispersive medium, wherein the balance between dispersion and nonlinearity effects enables the intensity to remain constant. In this manuscript, the dynamical behavior of Kudryashov’s equation is discussed with the assistance of truncated M-fractional derivative. Kudryashov’s equation is a mathematical model utilized to characterize complex phenomena in telecommunications and transmission technology. It describes the propagation of nonlinear pulses in optical fibers. Different forms of soliton solutions like bright, dark, singular and combo solitions have been extracted. Moreover, hyperbolic, periodic and Jacobi elliptic function solutions are recovered. Two recently modern integration tools like Φ6-expansion method and modified generalized exponential rational function method have been adopted to recover the solutions. The used methods not only provides previously extracted solutions but also secures new solutions. In order to visually depict the output, a variety of graphs featuring distinct shapes are produced in response to appropriate parameter values. The results obtained from this research indicate that the chosen methodologies are effective in improving the understanding of nonlinear dynamical phenomena. A large number of engineers who employ engineering models are expected to find this research interesting. The results demonstrate that the selected methods are practical, straightforward to implement, and applicable to intricate systems across numerous domains, with a specific emphasis on the field of optical fibers. The findings indicate that the system may contain a considerable number of soliton structures.

Solving the relativistic Toda lattice equation via the generalized exponential rational function method

Solving the relativistic Toda lattice equation via the generalized exponential rational function method

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

Optical Solitons for Nonlinear Schrodinger Equation Formatted in the Absence of Chromatic Dispersion Through Modified Exponential Rational Function Method and Other Distinct Schemes

Optical Solitons for Nonlinear Schrodinger Equation Formatted in the Absence of Chromatic Dispersion Through Modified Exponential Rational Function Method and Other Distinct Schemes

Bifurcation analysis, chaotic analysis and diverse optical soliton solutions of time-fractional (2+1) dimensional generalized Camassa-Holm Kadomtsev-Petviashvili equation arising in shallow water waves

This paper presents a study on (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation, which is used to describe the behavior of shallow water waves in nonlinear media. The considered equation provides a more accurate description of wave behavior compared to linear wave equations and can account for wave breaking and other nonlinear effects. This model can be used to describe and study the behavior of nonlinear waves such as rogue waves in complex fluid dynamics scenarios. This includes the behavior of waves in stratified fluids, nonlinear dispersive media and wave interactions in fluid flows with varying velocities and densities. The bifurcation analysis of the governing equation has been performed using the planar dynamical system method. The chaotic behavior of the dynamical system has been examined by utilizing various techniques such as time series analysis and the construction of 2D and 3D phase space trajectories. Furthermore, the introduction of a perturbed term has resulted in the observation of chaotic and quasi-periodic behaviors across a range of parameter values. The considered equation has been reduced to ordinary differential equation by performing symmetry reduction. The Kudryashov method has been used to obtain the exact solution of reduced equation. The single soliton solution of governed equation has been obtained by using Hirota method and impact of fractional parameter on the obtained solution has been studied using graphical representation. The extended sinh-Gordon equation expansion method and modified generalized exponential rational function method have been exploited to obtain dark, bright and singular soliton solutions of considered equation. The motivation for this study arises from the need to understand and analyze the complex dynamics of shallow water waves in nonlinear media with a particular focus on the (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation. By performing symmetry reduction and applying various analytical methods, we aim to unravel the intricate behavior and soliton solutions of considered equation, contributing to the broader understanding of nonlinear wave phenomena.

Structure of Analytical and Numerical Wave Solutions for the Nonlinear (1 + 1)-Coupled Drinfel’d–Sokolov–Wilson System Arising in Shallow Water Waves

In this article, we successfully obtain novel solutions for the coupled Drinfel’d–Sokolov–Wilson DSW system utilizing various methods. These include soliton solutions characterized by hyperbolic, rational, and trigonometric functions. Specifically, the generalized exponential rational function method (GERFM) and a modified version of the new Kudryashov method (MVNK) are employed to derive diverse soliton solutions for the system. Additionally, we demonstrate numerical solutions for the coupled Drinfel’d–Sokolov–Wilson system using adaptive moving mesh and uniform mesh methods. Also, we study the stability and error analysis of the numerical schemes. To validate the accuracy and reliability of the exact solutions obtained through analytical methods, we compare them with the numerical solutions both analytically and graphically. The techniques presented in this article are deemed suitable and acceptable and can be effectively applied to solve other nonlinear evolution systems.

New soliton solutions of kraenkel-manna-merle system with beta time derivative

This article discusses the fractional Kraenkel-Manna-Merle (KMM) system, which describes the motion of a nonlinear ultrashort wave pulse through saturated ferromagnetic materials with zero conductivity. The fractional behavior of this system was investigated using the beta derivative. The modified generalized exponential rational function method (MGERFM), developed by modifying the generalized exponential rational function method (GERFM), is applied to this system for the first time. Thus, some soliton solutions of the KMM system that have not been obtained before are presented for the first time in this study. In addition, 2D, 3D and density graphs of the obtained solutions for various values and ranges are presented. Discussions of these graphs are given and the found solutions are compared with other solutions.

Solitary wave solutions and investigation the effects of different wave velocities of the nonlinear modified Zakharov–Kuznetsov model for the wave propagation in nonlinear media

The modified Zakharov-Kuznetsov (mZK) model convey a significant role to analyze the inner mechanism of physical compound phenomenon in the field of two-dimensional discrete electrical lattice, the electrical waves in cold plasmas, plasma physics, nonlinear optics, wave behaviors of deep oceans, etc. In this study, numerous new, advanced, and more general exact solitary travelling wave solutions are explored to the early stated nonlinear model by the aid of newly developed generalized exponential rational function (GERF) method through the traveling wave transformation. The established solutions are in terms of trigonometric functions, rational functions, hyperbolic functions, exponential rational functions, and complex-soliton solutions. The results reveal that the free parameters significantly influence the existence of traveling waves, as well as nature, shape, and stability. The established soliton solutions demonstrate that the method is effective, compatible, scientifically efficient and easily applicable to identify a variety of wave structures of nonlinear evolution models. The mathematical simulations of the results were conducted by sketching 3D and 2D structures for different values of the associated parameters by the aid of the Wolfram Mathematica program.

On new diverse variety analytical optical soliton solutions to the perturbed nonlinear Schrödinger equation

This study focuses on investigating the optical soliton solutions for the perturbed nonlinear Schrödinger equation involving Kerr nonlinearity. It should be noted that the non-integrable nature of the nonlinear Schrödinger equation becomes apparent when the Kerr law nonlinearity is absent. This absence of integrability significantly obstructs finding exact solutions. Our research utilizes an innovative analytical technique, the modified generalized exponential rational function method, to derive new soliton solutions. The new soliton solutions we obtained are quite versatile and practical for real-world applications. They are expressed in terms of elementary functions such as exponentials, trigonometric functions, and hyperbolic functions. This simplicity allows for easy comprehension and straightforward utilization in various scenarios. We present a variety of visualizations and graphs to enhance the understanding of the physical behaviors exhibited by the new soliton solutions. The graphical outcomes give insights into soliton evolution and dynamics under various conditions. Our study shows the analytical technique’s efficacy in producing new soliton solutions for the perturbed nonlinear Schrödinger equation. We explore the possibility of expanding the method to discover solutions for alternative nonlinear partial differential equations that arise within soliton theory and nonlinear optics. The analytical approach offers a systematic framework for obtaining new soliton solutions for a broad class of nonlinear wave equations. Our employed methodologies hold also the potential to significantly advance the field and contribute to the development of new methodologies for tackling these challenging equations.

A study on the solutions of (3+1) conformal time derivative generalized q-deformed Sinh-Gordon equation

This article is about examining the solutions of the (3+1) conformal time derivative generalized q-deformed Sinh-Gordon equation. The integration method used to reach the solutions of the equation is the generalized exponential rational function method. In this article, the process of examining the solutions goes step by step, first the basic steps of the proposed method are given, then the reduction of the equation is examined, and then the solutions are obtained by applying the method. To perceive the physical phenomena, 2D and 3D graphical patterns of some of solutions obtained in this study are plotted by using computer programming. The worked-out solutions ascertained that the suggested method is effectual, simple and direct.

Optical solutions to a conformable fractional extended KdV model equation

In this paper, we intend to obtain the extended Korteweg-de Vries(K-dV) equation in the followingDταφ1+λ32[32λ4−(C2(1−f)+D2fσ2)]Dξ2αφ1+λ32(52λ6−C3(1−f)−fσ3D3)(Dξαφ1)3+λ32Dξ3αφ1=0by using the reductive perturbation approach and we explore solutions of this equation using the generalized exponential rational function method. The answers obtained in this study are new and have not been reported so far.

Wave Propagation and Stability Analysis for Ostrovsky and Symmetric Regularized Long-Wave Equations

This work focuses on the propagation of waves on the water’s surface, which can be described via different mathematical models. Here, we apply the generalized exponential rational function method (GERFM) to several nonlinear models of surface wave propagation to identify their multiple solitary wave structures. We provide stability analysis and graphical representations for the considered models. Additionally, this paper compares the results obtained in this work and existing solutions for the considered models in the literature. The effectiveness and potency of the utilized approach are demonstrated, indicating their applicability to a broad range of nonlinear partial differential equations in physical phenomena.

Investigate the dynamic nature of soliton solutions and bifurcation analysis to a new generalized two-dimensional nonlinear wave equation with its stability

In this current research, we investigate a new generalized two-dimensional nonlinear wave equation of engineering physics using two versatile approaches. Nonlinear wave models are very important in that they play a key role in modeling diverse occurrences in cosmology, solid mechanics and fluid dynamics, astrophysics, mathematical physics and engineering. The research work is based on three steps. Firstly, soliton solutions are secured via the modified Sardar sub-equation method (MSSEM) and the generalized exponential rational function method (GERFM). Secondly, for bifurcation analysis phase portraits of an equation are erected. Also, using the Galilean transformation, the given equation plane dynamical system is erected and all possible phase portraits are provided. Lastly, the modulation instability of the equation is examined. In order to better comprehend how the nonlinear wave equation depicts physical processes, the wave dynamics of the solutions are displayed to give the results a more physical perspective. The use of these findings in physics, engineering and complex media is essential.

Generalized exponential rational function method to the fractional shallow water wave phenomena

In this study we introduce the fractional shallow water wave phenomena, namely, the new extended (2 + 1)-dimensional fractional Boussinesq equation and in the following we apply the generalized exponential rational function for extracting the soliton solutions to the Ex-FBe (extended (2 + 1)-dimensional fractional Boussinesq equation) by using Maple tools for calculating and Mathematica for graphical behaviors. Many types of exact soliton solutions for the fswwp equation have been derived by the erf method. Physical interpretations of some of the obtained solutions are also included. It can be easily concluded that the new proposed method is very efficient and finds the exact solutions of the equation in a relatively easy way.