Simple Equation Method Research Articles

Controlling Soliton Pulse Propagation in Inhomogeneous Optical Waveguides With Dual‐Power Law Refractive Index

ABSTRACTThis study focuses on the manipulation of soliton parameters within optical waveguides characterized by a dual‐power law refractive index, drawing similarities with published papers. The precise management of solitons holds significant importance in the field of fiber‐optical communication. While previous research has primarily concentrated on modifying soliton attributes such as amplitude and width, comparatively less attention has been dedicated to controlling the wave speed of solitons. In order to bridge this gap, the researchers adopted the employment of the nonlinear Schrödinger equation with time‐dependent dispersion and two power‐law nonlinearities, mirroring similar approaches found in published literature. By employing this methodology, the researchers successfully achieved active control over the wave speed of solitons in non‐uniform optical waveguides. To effectively handle the inherent complexities of the nonlinear system, two well‐established techniques, namely, the generalized Kudryashov method and the simple equation method, were utilized to derive solutions. These techniques have been previously employed and demonstrated in published papers, further enhancing the validity and reliability of the research findings. We discuss the traveling wave solutions by finding the fixed points of the dynamical planar system.

Exploring the Diversity of Kink Solitons in (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves.

A diverse array of optical solitons of generalized nonlinear dispersive mK(m,n) model

The generalized [Formula: see text] model is used to portray the movement of nonlinear waves by applying four mathematical methods with assistance of Mathematica software. These methods are called enhanced simple equation (SE) method, Exp[Formula: see text]-expansion method, [Formula: see text]-expansion method and modified F-expansion method. For physical behavior, few solutions are plotted graphically in the form of two-dimensional (2D) and three-dimensional (3D). The techniques and soliton solutions that are developed provide computational tools for additional research in this area. With this work, we have gained a better knowledge of soliton motion and how it is used in dynamical systems and other related technologies. Hence, this work will help physicists envision some new ideas and hypotheses in nonlinear science.

Periodic and Axial Perturbations of Chaotic Solitons in the Realm of Complex Structured Quintic Swift-Hohenberg Equation

This research work employs a powerful analytical method known as the Riccati Modified Extended Simple Equation Method (RMESEM) to investigate and analyse chaotic soliton solutions of the (1 + 1)-dimensional Complex Quintic Swift–Hohenberg Equation (CQSHE). This model serves to describe complex dissipative systems that produce patterns. We have found that there exist numerous chaotic soliton solutions with periodic and axial perturbations to the intended CQSHE, provided that the coefficients are constrained by certain conditions. Furthermore, by applying a sophisticated transformation, the provided transformative approach RMESEM transforms CQSHE into a set of Nonlinear Ordinary Differential Equations (NODEs). The resulting set of NODEs is then transformed into an algebraic system of equations by incorporating the extended Riccati NODE to assume a series form solution. The soliton solutions to this system of equations can be found as periodic, hyperbolic, exponential, rational-hyperbolic, and rational families of functions. A variety of 3D and contour visuals are also provided to graphically illustrate the axially and periodically perturbed dynamics of these chaotic soliton solutions and the formation of fractals. Our findings are noteworthy because they shed light on the chaotic nature of the framework we are examining, enabling us to better understand the dynamics that underlie it.

Exploring the dynamical behaviour of optical solitons in integrable kairat-II and kairat-X equations

The current research focusses on the establishment of an analytical approach known as the Riccati Modified Extended Simple equation Method (RMESEM) for the development and assessment of optical soliton solutions in two important Kairat equations. These models are known as Kairat-X equation (K-XE) and the Kairat-II equation (K-IIE), which describe the trajectory of optical pulses in optical fibres. Using RMESEM, the soliton solutions in five families–the periodic, rational, hyperbolic, rational-hyperbolic, and exponential functional families–are achieved for the targeted models. A set of 3D, 2D, and contour visualisations are presented to visually illustrate the dynamics of some produced optical soliton solutions which demonstrates that the due to the axial-periodic perturbation, the optical soliton solutions exhibit fractal phenomena in the realm of K-IIE whereas in the setting of K-XE the optical solitons adopt the form of kink solitons such as solitary kink, lump-type kink, dromion and periodic kink soliton structures. Moreover, our suggested RMESEM illustrates its usefulness by building a multitude of optical soliton solutions, providing valuable insights into the dynamics of the targeted models and indicating potential uses in addressing other nonlinear models.

On the study of an extended coupled KdV system: Analytical solutions and conservation laws

This paper aims to derive analytical solutions of an extended (2+1)-dimensional constant coefficients new coupled Korteweg–de Vries system. This will be achieved by implementing the classical symmetry method in conjunction with some simplest equation methods. The simplest equations that will be utilised includes among others the Bernoulli and Riccati equations. Furthermore, the conservation laws will be constructed through the multipliers approach, which subsequently reveals the conserved quantities. Moreover, a brief presentation of results obtained consisting of a variety of profile structures which include the kink type, bell and inverted bell shaped and singular wave solutions will be discussed.

Analyzing the dynamics of multi-solitons and other solitons in the perturbed nonlinear Schrödinger equation

The perturbed nonlinear Schrödinger equation is employed to characterize the dynamics of optical wave propagation when confronted with dissipation (or gain) and nonlinear dispersion that vary with both time and space. This equation serves as a fundamental model for investigating pulse dynamics within optical fibers and has application to nanofiber applications. This study successfully discovers optical solitons within this framework using the unified solver, Jacobi elliptic function, and simplest equation methods. We extract solutions using hyperbolic, trigonometric, and rational functions, including multi-solitons, dark, singular, bright, and periodic singular solitons. This study thoroughly compares our results with existing literature to provide novelty and significance of our findings. We have incorporated a detailed comparison between the methods employed in our study, which highlights their importance and strength. We have derived soliton solutions for the examined equations and generated 3D contour and 2D visual representations of the resulting solution functions. Alongside obtaining the soliton solutions, we offer a graphical exploration of how the parameters in the considered equations influence the system.

Travelling wave solutions and conservation laws of the (2+1)-dimensional new generalized Korteweg–de Vries equation

In this study, we investigate the travelling wave solutions of the (2+1)-dimensional new generalized Korteweg–de Vries equation by employing Lie group analysis along with various techniques which include direct integration, simplest equation method and Kudryashov’s method. The results obtained consists of periodic, kink, soliton and hyperbolic solutions. The symbolic computation software Maple is used to check the accuracy of all solutions obtained. Finally, 3D, density, and 2D plots of the derived solution are displayed to show the physical appearance of the model. Furthermore, we utilize the general multiplier technique and Ibragimov’s method to derive its conserved vectors. Conservation of energy and momentum, amongst others were found. Conservation laws have many significant uses with regards to integrability, linearization and analysis of solutions.

Assorted Spatial Optical Dynamics of a Generalized Fractional Quadruple Nematic Liquid Crystal System in Non-Local Media

Nematicons upgrade the recognition of light localization in the reorientation of non-local media. the current research employs a powerful integral scheme using a different procedure, namely, the modified simple equation method (MSEM), to analyze nematicons in liquid crystals from the controlling model. The expanded MSEM is investigated to enlarge the applicability of the standard one. The suggested expansion depends on merging the MSEM and the ansatz method. The new generalized nonlinear n-times quadruple power law is included. With the aid of the symbolic computational package Mathematica, new explicit complex hyperbolic, periodic, and more exact spatial soliton solutions are derived. Moreover, the related existence constraints are obtained. To show the dynamical properties of some of the obtained nematicons, three-dimensional profiles with corresponding contours are depicted with the choice of appropriate values of arbitrary parameters. The fractional impacts in various applicable senses are analyzed to investigate the generality of the considered model.

New computational approaches to the fractional coupled nonlinear Helmholtz equation

PurposeThe main aim of this paper is to investigate the fractional coupled nonlinear Helmholtz equation by two new analytical methods.Design/methodology/approachThis article takes an inaugural look at the fractional coupled nonlinear Helmholtz equation by using the conformable derivative. It successfully finds new fractional periodic solutions and solitary wave solutions by employing methods such as the fractional method and the fractional simple equation method. The dynamics of these fractional periodic solutions and solitary wave solutions are then graphically represented in 3D with appropriate parameters and fractal dimensions. This research contributes to a deeper comprehension and detailed exploration of the dynamics involved in high dimensional solitary wave propagation.FindingsThe proposed two mathematical approaches are simple and efficient to solve fractional evolution equations.Originality/valueThe fractional coupled nonlinear Helmholtz equation is described by using the conformable derivative for the first time. The obtained fractional periodic solutions and solitary wave solutions are completely new.

Modulation instability analysis, and characterize time-dependent variable coefficient solutions in electromagnetic transmission and biological field

This study integrates the telegraph model with the traveling wave coefficient. This model characterizes planar random motion of particles in fluid flow, pressure waves from pulsatile blood flow in arteries, electrical impulses in nerve and muscle cell axons, and electromagnetic waves in superconducting media. Using the efficient and well-established Enhanced Modified Simple Equation (EMSE) method, we investigate solitary wave solution with the variable coefficient of the telegraph model. To investigate the variable coefficient solitary wave solution, we apply a transformation variable η = h(t)x + g(t), which is dependent on the time variable function. The diverse functions of h(t) and g(t) yield many novel and exclusive solutions. Numerical solutions are plotted in 3D, density, and 2D. instanton soliton, kinky periodic wave, lump wave, kink wave, anti-kink wave, double periodic wave, diverse types periodic wave, lump with bell wave periodic wave, periodic lump wave, etc. Solitary waves explain complex wave phenomena through nonlinear effects like self-interaction and dispersion. Due to the classical nonlinear telegraph model's wave dynamics, they are used in brain function and communication system design. We conclude by testing the governing model's modulation instability. Dispersive and nonlinear processes modulating the stable state cause instability in high-order nonlinear equations. Examining the wave movement role and modulation instability analysis to assess solution stability shows that all solutions are accurate and stable. The computational difficulties and results demonstrate the approaches' clarity, effectiveness, and simplicity, suggesting they can be applied to evolutionary dynamic and static nonlinear equations in computational physics, other real-world situations, and numerous academic disciplines.

Constructing the soliton wave structure to the nonlinear fractional Kairat-X dynamical equation under computational approach

In this paper, the nonlinear fractional Kairat-X equation is investigated on the basis of computational simulation. The nonlinear fractional Kairat-X equation is an integrable equation and is used to explain the differential geometry of curves and equivalence aspects. Several kinds of solitary wave structures of the nonlinear fractional Kairat-X equation are established successfully via the implantation of the extended simple equation method. Here, we explore the interesting, novel and general solutions in trigonometric, exponential, and rational types, which represent periodic wave solitons, mixed solitons in the shape of bright–dark solutions, kink wave solitons, peakon bright and dark solitons, anti-kink wave solutions, bright solitons, dark solitons, and solitary wave structure. The physical structures of secured results, aided by numerical simulation, have numerous applications in applied sciences such as optical fiber, geophysics, laser optics, mathematical physics, nonlinear optics, nonlinear dynamics, communication system, and engineering. This study explores the physical behavior of models through the visualization of solutions in contour, 2D and 3D plots by revealing that these solutions yield profitable results in the field of mathematical physics. The study demonstrates that the proposed technique is more reliable, efficient, and powerful in analyzing nonlinear evolution equations in various domains of science.

On some novel solitonic structures for the Zhiber–Shabat model in modern physics

Abstract In this article, the modified Kudryashov and extended simple equation methods are employed to obtain analytical solutions for the Zhiber–Shabat problem. The outcomes of this study clearly indicate that the provided methodologies are appropriate techniques for generating some new exact solutions for nonlinear evolution equations. Furthermore, the nature of the solutions would be presented in three dimensions for various parameters applying the most advanced scientific instruments. The physical behavior of the solutions are graphically displayed, and it is established that the acquired solutions are newly constructed in the form of bright, dark, optical, singular, and bell-shaped periodic soliton wave structures. The properties of the nonlinear model have been illustrated using 3D, 2D, and contour plots by selecting an appropriate set of parameters, which is demonstrated to visualize the physical structures more productively. Finally, it is concluded that similar strategies can also be implemented to study many contemporary models. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the governing model appearing in diverse disciplines. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation (LGHe), which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves. The LGHe finds applications in various scientific fields, including fluid dynamics, plasma physics, biological systems, and electricity-electronics. The study adopts Lie symmetry analysis as the primary framework for exploration. This analysis involves the identification of Lie point symmetries that are admitted by the differential equation. By leveraging these Lie point symmetries, symmetry reductions are performed, leading to the discovery of group invariant solutions. To obtain explicit solutions, several mathematical methods are applied, including Kudryashov’s method, the extended Jacobi elliptic function expansion method, the power series method, and the simplest equation method. These methods yield solutions characterized by exponential, hyperbolic, and elliptic functions. The obtained solutions are visually represented through 3D, 2D, and density plots, which effectively illustrate the nature of the solutions. These plots depict various patterns, such as kink-shaped, singular kink-shaped, bell-shaped, and periodic solutions. Finally, the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a crucial role in the study of physical quantities, such as the conservation of energy and momentum, and contribute to the understanding of the underlying physics of the system.

Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u=be−ax2, a<0, a,b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u=be−ax22, respectively, for the third-order PDE, u=AeiΦ, where the amplitude b and the phase function a are complex-valued functions, A>0, and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.

Optimal Choice of the Auxiliary Equation for Finding Symmetric Solutions of Reaction–Diffusion Equations

This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two a priori choices: a power series in which solutions are sought and a predefined auxiliary equation. Uninspired choices can block the solving process. We propose a procedure that allows for the establishment of their optimal forms, compatible with the nonlinear equation to be solved. The procedure will be illustrated on the rather large class of reaction–diffusion equations, with examples of two of its subclasses: those containing the Chafee–Infante and Dodd–Bullough–Mikhailov models, respectively. We will see that Riccati is the optimal auxiliary equation for solving the first model, while it cannot directly solve the second. The elliptic Jacobi equation represents the most natural and suitable choice in this second case.

Exact wave solutions of new generalized Bogoyavlensky–Konopelchenko model in fluid mechanics

In this paper, we examine a novel generalized [Formula: see text]-dimensional Bogoyavlensky–Konopelchenko (gBK) model analytically via application of five mathematical methods, namely called Extended Simple Equation (ESE) method, Modified Extended Auxiliary equation mapping (MEAEM) method, [Formula: see text]-expansion method, [Formula: see text]-expansion method and modified F-expansion method, respectively, to obtain different types solitary wave solutions in the form of trigonometric, hyperbolic, rational and exponential functions as compared to solutions exist in [S. T. Chen and W. X. Ma, Math. China 13 (2018) 525; A. Sonia, A. Jamshad, S.-U. Rehman and A. Ali, Int. J. Appl. Comput. Math. 9 (2023)]. The research focuses on physical systems, physiology, mathematical applications, engineering fields, and the chemical processes of species in the porous nanoparticles, examining the impact of various parameters on the nonlinear model. The graphs of some solutions are plotted in the form of two-dimensional and three-dimensional by assigning the particular values to the paraments with the assistance of mathematica software which illustrated the physical behavior of the concern model. The derived solution has numerous applications in fluid dynamics and the interaction of Riemann waves and long waves.

Exploration of unexpected optical mixed, singular, periodic and other soliton structure to the complex nonlinear Kuralay-IIA equation

In this research work, we explored novel soliton solutions to the complex nonlinear Kuralay-II (K-IIA) equation on the bases of computational simulation. This complex nonlinear equation is used in many fields such as, optical fibers, ferromagnetic materials and nonlinear optics. We utilized the extended simple equation method on complex nonlinear Kuralay-IIA equation and extracted the unexpected soliton solutions. The extracted soliton solutions having various kinds of physical structure such as bright solitons, periodic wave solitons, kink wave solitons, peakon solitons, dark solitons, anti-kink wave solitons, mixed bright and dark solitons, periodic travelling waves, singular solitons and solitary waves. The graphically analysis of obtained solutions demonstrated with absolute, real and imaginary values of the function visualizing in three dimensional, two dimensional and contour graphics under the numerical simulation by utilizing the Mathematica software. The extracted soliton solutions in this research provide valuable insights into this complex nonlinear equation. The results offer a framework for further analysis, making the solutions effective, easy to apply, and reliable for future complex nonlinear problems development, making them valuable for future. The method used in this work effective, reliable, powerful, easy to apply for other nonlinear partial differential equations.

New Wave Behaviors Generated by Simple Equation Method with Riccati Equation of Some Fourth-Order Fractional Water Wave Equations

New Wave Behaviors Generated by Simple Equation Method with Riccati Equation of Some Fourth-Order Fractional Water Wave Equations

IMPLEMENTATION AND ASSESSMENT OF THE SIMPLE EQUATION TECHNIQUE FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

In this paper, the simple equation method is especially used to solve two Nonlinear Partial Differential Partial Equations NLPDEs, the Kodomstev-Petviashvili (KP) equation and the (2+1)-dimensional breaking soliton equation. The modified Benjamin-Bona-Mahony equation and the Klein-Gordon equation in (1+2) dimensions are two illustrations of second order nonlinear equations that can benefit from using this approach. The Bernoulli equation acts as the trial condition and aids in the mathematical a description of the nonlinear wave equation in the simple equation.