Dimensional Zoomeron Research Articles

Obtaining analytical solutions of (2+1)-dimensional nonlinear Zoomeron equation by using modified F-expansion and modified generalized Kudryashov methods

PurposeThe purpose of the article is to conduct a mathematical and theoretical analysis of soliton solutions for a specific nonlinear evolution equation known as the (2 + 1)-dimensional Zoomeron equation. Solitons are solitary wave solutions that maintain their shape and propagate without changing form in certain nonlinear wave equations. The Zoomeron equation appears to be a special model in this context and is associated with other types of solitons, such as Boomeron and Trappon solitons. In this work, the authors employ two mathematical methods, the modified F-expansion approach with the Riccati equation and the modified generalized Kudryashov’s methods, to derive various types of soliton solutions. These solutions include kink solitons, dark solitons, bright solitons, singular solitons, periodic singular solitons and rational solitons. The authors also present these solutions in different dimensions, including two-dimensional, three-dimensional and contour graphics, which can help visualize and understand the behavior of these solitons in the context of the Zoomeron equation. The primary goal of this article is to contribute to the understanding of soliton solutions in the context of the (2 + 1)-dimensional Zoomeron equation, and it serves as a mathematical and theoretical exploration of the properties and characteristics of these solitons in this specific nonlinear wave equation.Design/methodology/approachThe article’s methodology involves applying specialized mathematical techniques to analyze and derive soliton solutions for the (2 + 1)-dimensional Zoomeron equation and then presenting these solutions graphically. The overall goal is to contribute to the understanding of soliton behavior in this specific nonlinear equation and potentially uncover new insights or applications of these soliton solutions.FindingsAs for the findings of the article, they can be summarized as follows: The article provides a systematic exploration of the (2 + 1)-dimensional Zoomeron equation and its soliton solutions, which include different types of solitons. The key findings of the article are likely to include the derivation of exact mathematical expressions that describe these solitons and the successful visualization of these solutions. These findings contribute to a better understanding of solitons in this specific nonlinear wave equation, potentially shedding light on their behavior and applications within the context of the Zoomeron equation.Originality/valueThe originality of this article is rooted in its exploration of soliton solutions within the (2 + 1)-dimensional Zoomeron equation, its application of specialized mathematical methods and its successful presentation of various soliton types through graphical representations. This research adds to the understanding of solitons in this specific nonlinear equation and potentially offers new insights and applications in this field.

Exact solutions of the (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional Zoomeron model arising in nonlinear optics via mapping method

The Zoomeron model covers particular kinds of solitons with distinctive properties that appear in several physical scenarios, such as, fluid dynamics, nonlinear optics and laser physics. First time utilising the mapping method, we determine the analytical solution to the described model, including several novel dynamical behaviours. Through symbolic computation, we are able to derive the breather waves, kink waves, dark soliton, singular soliton, periodic soliton and bright soliton of this model. Additionally, we encounter single kink waves and single breather waves. We find novel hyperbolic trigonometric, rational and elliptic functions. Modelling our observations with MATLAB tools and producing many 3D graphs. The results obtained will be crucial for further research on complicated nonlinear models.

Analytical solution of the (2+1)-dimensional Zoomeron equation by rational sine-Gordon Method

The current study is about the solution of the Zoomeron equation, one of the important models of mathematics and physics. In this study, the rational Sine-Gordon expansion method (RSGEM) is used to obtain various analytical solutions of the model. Compared to other methods, this method is quite effective and the desired results were obtained. Although there are many analytical solutions to the model used in the literature, we present rational type solutions for the first time with this method. We obtained rational hyperbolic function solutions, and also classified all soliton solutions (kink-like, kink, singular kink, anti-kink, dark, bright). In addition, geometric representations of the solutions in two-, and three-dimensional space and contour shape are made with the Mathematica software program.

Study of explicit travelling wave solutions of nonlinear (2 + 1)-dimensional Zoomeron model in mathematical physics

Abstract This article studeis the nonlinear (2 + 1)-dimensional Zoomeron equation by utilizing the various prominent analytical approaches namely the unified method and the extended hyperbolic function approach. The analysis in the current paper demonstrates the presence of travelling wave solutions. The applied methods are utilized as powerful tools to investigate and solve the model. The results obtained through these analytical methods reveal insightful patterns in the behavior of the Zoomeron equation. The significance of our work lies in the uniqueness of the methods employed. The two methods are applied to systematically analyze the equation, revealing hidden patterns and structures within its solution space. This leads to the discovery of a collection of solitary wave solutions such as kink waves, singular kink waves, periodic waves and dark soliton using contour plots, 3D and 2D graphics. In this article, we definitely prove that as the free parameters change, the wave amplitude changes as well. It is shown that the applied strategies are more effective and may be implemented to a variety of contemporary nonlinear evolution models emerging in mathematical physics.

New wave behaviors and stability analysis for the (2+1)-dimensional Zoomeron model

New wave behaviors and stability analysis for the (2+1)-dimensional Zoomeron model

Dynamics Investigation and Solitons Formation for (2+1) -Dimensional Zoomeron Equation and Foam Drainage Equation

The prime goal of this study is to investigate novel solutions of two well-known nonlinear models namely the (2+1)-dimensional Zoomeron equation and the foam drainage equation by utilizing a powerful technique; the extended left(frac{G'}{G^{2}}right)-expansion method. Using this methodology, the hyperbolic function solutions, the trigonometric function solutions and the rational function solutions are constructed. Abundant soliton solutions are retrieved from the obtained results. The dynamical structures of the solutions are illustrated graphically through 3-dimensional graphs and the corresponding contour plots. The reported results depict the effectiveness and capability of the extended left(frac{G'}{G^{2}}right)-expansion method for handling different nonlinear partial differential equations.

Study of explicit travelling wave solutions of nonlinear evolution equations

The Bernoulli Sub-ODE approach is used in this study to look for comprehensive travelling wave solutions to the nonlinear evolution equations (NLEEs). The analysis in the present paper shows the existence of travelling waves for the time-regularized long-wave (TRLW) equation, the modified Korteweg–de Vries –Zakharov–Kuznetsov (mKdV–ZK) equation, and the (2+1)-dimensional Zoomeron equation. The outcomes demonstrate the richness of explicit solutions of the studied models. As a result, precise solitary wave solutions to the studied problems, such as kink waves, singular kink waves, dark soliton, and periodic waves are found. The phase plane is briefly examined after the determination of the Hamiltonian function. Using Maple 13, we validated the accuracy of the obtained solutions by reintroducing them into the original equation. We will demonstrate how the amplitudes and wave profiles are impacted by free parameters. In this article, we firmly establish that the wave amplitude varies as the free parameters change. It is demonstrated that the technique is efficient and applicable to several different NLEEs in mathematical physics.

An encyclopedia of Kudryashov’s integrability approaches applicable to optoelectronic devices

Purpose:In the last decades, many researchers have performed their best in order to find the solution of the nonlinear evolution equations, nonlinear partial differential equations (NLPDEs), fractional nonlinear partial differential equations and nonlinear optical models by using Kudryashov methods, which have various forms. However, most researchers have used different methods under the same name or applied the same method under the different names. This situation has created confusion for researchers and readers, such as choosing the appropriate method, algorithm and the right article title. In order to eliminate this problem and to clarify the methods, we have conducted rigorous research and wrote a certain standard in this field by bringing together the Kudryashov methods, which are widely used under different names in the literature. Besides, it has been aimed to give the general algorithm of the methods, to make some addendums on the methods and to present new approaches to these methods for the researchers. We strongly believe that the study will be of great benefit not only for those who are interested in nonlinear evolution problems, but also for studies in the field optics. Methodology:Regarding the methods, first of all, the articles of the researchers who suggested the methods and the studies of the respected researchers in this field have been examined and tried to gather under one framework and give a general name with appropriate algorithm for each Kudryashov method. The algorithms and solution functions of each method have been checked on the related Riccati and Bernoulli equations with different softwares (Mathematica, Maple, Matlab, etc.), and confirmed every results carefully. Afterwards, all methods have been applied on (2+1)-dimensional Zoomeron equation for the global test purpose. With the usual standard approach, the nonlinear partial differential form of Zoomeron equation has been transformed into the nonlinear ordinary differential (NODE) form by applying the wave transformation, then each method applied to the NODE form of Zoomeron equation. After these operations, we have obtained a system of linear algebraic equations and their appropriate solutions. By obtaining the appropriate solution sets, the soliton solutions of the Zoomeron equation have been constructed and some selected graphical presentations given. Findings:The results have been obtained from the applications that applied on the Zoomeron equation via Kudryashov methods shows us they can be used widely and effectively in the solution of not only nonlinear evolution problems, but also highly dispersive and higher order nonlinear partial differential equations. It has been seen that all the applied Kudryashov methods give effective results. Besides, controlling and using the parameters in the Kudryashov methods play important role, especially obtaining the effective and desired solution sets. Originality:Based on our detailed research, we could not find any summative study in this area like this one. In addition to all these, some of the approaches and addendums have been presented for the first time in this paper. Besides, (2+1)-dimensional Zoomeron equation has also not been previously solved by some of these methods. It is thought that this study will be used as a reference by many researchers in the future and it will be of great benefit to researchers.

Computational wave solutions of some nonlinear evolution equations

This paper aims to realize some of the vital nonlinear evolution of partial differential equations in mathematical physics. We implement an extended simple equation (ESE) method to get exact and solitary traveling wave solutions of the Calogero-Degasperis (CD) equation, potential Kadomtsev–Petviashvili (pKP) equation, generalized Korteweg–de Vries (gKdV) equation, and the (2 + 1)–dimensional Zoomeron (Z) equation. We obtain a new form of solutions for each model as mentioned above. This method is considered a new, robust, practical, and direct method. As we see, the remarkable ability of this method to apply on the fractional and integer order of nonlinear partial differential equations.

An investigation of the physical dynamics of a traveling wave solution called a bright soliton

This study examines the -dimensional Zoomeron equation, which has recently become popular in applied mathematics and physics. Bright soliton (non-topological), kink wave solution and traveling wave solutions are generated with the advantages of the generalized exponential rational function method. With the help of this method, it is aimed to produce different types of solutions for the Zoomeron equation compared to other traditional exponential function methods. The effects of parameters on the amplitude of the wave function are discussed, along with physical explanations backed by simulations. In addition, the advantages and disadvantages of the method for the -dimensional Zoomeron equation are discussed.

Diverse novel computational wave solutions of the time fractional Kolmogorov—Petrovskii - Piskunov and the (2 + 1)-dimensional Zoomeron equations

The numerical wave solutions of two fractional biomathematical and statistical physics models (the Kolmogorov—Petrovskii - Piskunov (KPP) equation and the (2 + 1)-dimensional Zoomeron (Z) equation) are investigated in this manuscript. Many novel analytical solutions in different mathematical formulations such as trigonometric, hyperbolic, exponential, and so on can be constructed using the generalized Riccati—expansion analytical scheme and the Caputo—Fabrizio fractional derivative. The fractional nonlinear evolution equation is converted into an ordinary differential equation with an integer order using this fractional operator. The obtained solution is used to describe the transmission of a preferred allele and the nonlinear interaction of moving waves, and the relative wave mode’s amplitude dynamic. To illustrate the fractional examined models, several drawings are explained in two dimensions and density plots.

Exact Solutions of Two Nonlinear Partial Differential Equations by the First Integral Method

In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software; the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.

Exact Solutions for nonlinear partial differential equation by modified F-expansion method

In this Paper, we use modified F-expansion method to construct new exact traveling wave solutions of nonlinear partial differential equation, The (2+1) – dimensional zoomeron equation, The obtained solutions include soliton – like solutions, trigonometric function solutions, rational solutions and exponential solutions.

Symmetry Analysis and Conservation Laws of the Zoomeron Equation

In this work, we study the (2 + 1)-dimensional Zoomeron equation which is an extension of the famous (1 + 1)-dimensional Zoomeron equation that has been studied extensively in the literature. Using classical Lie point symmetries admitted by the equation, for the first time we develop an optimal system of one-dimensional subalgebras. Based on this optimal system, we obtain symmetry reductions and new group-invariant solutions. Again for the first time, we construct the conservation laws of the underlying equation using the multiplier method.

Study of the Analytical Treatment of the (2+1)-Dimensional Zoomeron, the Duffing and the SRLW Equations via a New Analytical Approach

In this paper, we applied the improved tan (�(ξ )/ 2)-expansion scheme for the (2+1)-dimensional Zoomeron, the Duffing and the symmetric regularized long wave equa- tionsandexactparticularsolutionshavebeenfound.Theexactparticularsolutionscontaining four types hyperbolic function solution, trigonometric function solution, exponential solu- tion and rational solution. We obtained the further solutions comparing with other methods as sine-cosine function method (Qawasmeh in J Math Comput Sci 3:1475-1480, 2013). Recently this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that this method, with the help of symbolic com- putation, provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.

Bounded Traveling Waves of the (2+1)-Dimensional Zoomeron Equation

The bifurcation method of dynamical system and numerical simulation method of differential equation are employed to investigate the (2+1)-dimensional Zoomeron equation. We obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits. According to these phase portraits, all bounded traveling waves are identified and simulated, including solitary wave solutions, shock wave solutions, and periodic wave solutions. Furthermore, all exact expressions of these bounded traveling waves are given. Among them, the elliptic function periodic wave solutions are new solutions.

A note on enhanced (G′/G)-expansion method in nonlinear physics

In this talk we have applied an enhanced (G′/G)-expansion method to find the traveling wave solutions of the (2+1)-dimensional Zoomeron equation. The efficiency of this method for finding the exact solutions has been demonstrated. As a result, a set of exact solutions are derived, which can be expressed by the hyperbolic and trigonometric functions involving several parameters. When these parameters are taken as special values, the solitary wave solutions and the periodic wave solutions have been originated from the exact solutions. It has been shown that this method is effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics.

Symbolic Computation and New Exact Travelling Solutions for the (2+1)-Dimensional Zoomeron Equation

In this paper, we present Yan’s sine-cosine method and Wazwaz’s sine-cosine method to solve the (2+1)-dimensional Zoomeron equation. New exact travelling wave solutions are explicitly obtained with the aid of symbolic computation. The study confirms the power of the two schemes.

Traveling wave solutions of the (2 + 1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method

The modified simple equation (MSE) method is promising for finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this letter, we investigate solutions of the (2+1)-dimensional Zoomeron equation and the (2+1)-dimensional Burgers equation by using the MSE method and the Exp-function method. The competence of the methods for constructing exact solutions has been established.

Mathematical methods for a reliable treatment of the (2+1)-dimensional Zoomeron equation

Purpose: This paper investigates an analytical solution to a physical model called (2 + 1)-dimensional Zoomeron equation. Methods: The solutions of Zoomeron are obtained using direct methods such as the extended tanh, the exponential function and the sech p − tanh p function methods. Results: Several soliton solutions are obtained using the proposed methods. Conclusions: The obtained solutions are new, and each has its own structure.