Bogoyavlenskii Equation Research Articles

On Lie symmetries and soliton solutions of $$(2+1)$$ ( 2 + 1 ) -dimensional Bogoyavlenskii equations

The present article is devoted to find some invariant solutions of the $$(2+1)$$ -dimensional Bogoyavlenskii equations using similarity transformations method. The system describes $$(2+1)$$ -dimensional interaction of a Riemann wave propagating along y-axis with a long wave along x-axis. All possible vector fields, commutative relations and symmetry reductions are obtained by using invariance property of Lie group. Meanwhile, the method reduces the number of independent variables by one, which leads to the reduction of Bogoyavlenskii equations into a system of ordinary differential equations. The system so obtained is solved under some parametric restrictions and provides invariant solutions. The derived solutions are much efficient to explain the several physical properties depending upon various existing arbitrary constants and functions. Moreover, some of them are more general than previously established results (Peng and Shen in Pramana 67:449–456, 2006; Malik et al. in Comput Math Appl 64:2850–2859, 2012; Zahran and Khater in Appl Math Model 40:1769–1775, 2016; Zayed and Al-Nowehy in Opt Quant Electron 49(359):1–23, 2017). In order to provide rich physical structures, the solutions are supplemented by numerical simulation, which yield some positons, negatons, kinks, wavefront, multisoliton and asymptotic nature.

New Bäcklund transformations of the (2+1)-dimensional Bogoyavlenskii equation via localization of residual symmetries

The residual symmetry of the (2+1)-dimensional Bogoyavlenskii equation is obtained and localized to a Lie point symmetry by introducing new dependent variables to enlarge the system, then the corresponding finite transformation is obtained by using Lie’s first theorem. Furthermore, by introducing more dependent variables, the linear superposition of arbitrary number of residual symmetries is also localized and the corresponding finite transformations which are just Nth Bäcklund transformations of the (2+1)-dimensional Bogoyavlenskii equation are obtained in determinants form with some concrete solutions explicitly given.

Exact Solutions and Conservation Laws of the Bogoyavlenskii Equation

Exact Solutions and Conservation Laws of the Bogoyavlenskii Equation

Multi-soliton solutions of the generalized variable-coefficient Bogoyavlenskii equation

ABSTRACTIn this paper, generalized variable-coefficient Bogoyavlenskii equation is investigated. Via the dependent variable transformations, bilinear forms and multi-soliton solutions of the Bogoyavlenskii equation are obtained. Soliton propagation and interaction are analyzed: parameter affects the periodic/nonperiodic variety of the group velocity. Asymptotic analysis and Linear stability analysis are both analyzed.

Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

AbstractWe study here the initial value problem for a two‐dimensional Korteweg–de Vries (KdV) equation, first derived by Calogero and Bogoyavlenskii, by means of the inverse scattering transform. The dynamics of the discrete spectrum of an associated Schrödinger operator is far richer than that of KdV equation. Even for optimal eigenvalues, generic smooth solutions may develop shocks with multiple branches and/or cusp singularities in finite time. However, evolution may move poles of the transmission coefficient off the imaginary axis, destroy or even create them. We characterize conditions to prevent these pathologies before explosion time and describe ample classes of solutions, corresponding to both continuous and discrete spectrum. We also find that in certain conditions new eigenvalues might be created; in these cases a minimal set of initial spectral data must incorporate additionally the transmission coefficient on the entire plane. The previous results are applied to describe the Cauchy problem corresponding to initial data combinations of delta terms and derivatives and show that for long time the delta singularity may persist or be smoothed to a cusp‐discontinuity. Finally, we give conditions under which the evolution is reduced to the classical KdV.

The first integral method applied to the Bogoyavlenskii equations by means of conformable fractional derivative

In this paper, we construct exact solutions for the space–time nonlinear conformable fractional Bogoyavlenskii equations by using the first integral method, and with the help of Maple. As a result, generalized hyperbolic function solutions, generalized trigonometric function solutions and rational function solutions with free and deformation parameters are obtained. The method is very suitable, easy and effective handling of the solution process of nonlinear conformable fractional equations.

Solitons and other solutions to the nonlinear Bogoyavlenskii equations using the generalized Riccati equation mapping method

Recently, the authors of the article (Zahran and khater in Appl Math Model 40:1769–1775, 2016) have applied the modified extended tanh-function method to the nonlinear Bogoyavlenskii equations and found very few special solutions. In our article, we generalize this work by showing that the modified extended tanh-function method is just a special case of the generalized Riccati equation mapping method. Many families of exact solutions of the nonlinear Bogoyavlenskii equations have been found using the generalized Riccati equation mapping method. Bright–dark–singular soliton solutions and other solutions are obtained. Comparing our results with the well-known results are given.

Application of $$\frac{G'}{G^2}$$ G ′ G 2 -Expansion Method for Solving Fractional Differential Equations

In this paper, we apply a fractional \(\frac{G'}{G^2}\)-expansion method to look exact solutions of nonlinear fractional differential equations. A fractional complex transformation is used to convert a nonlinear FDE with Jumarie modified Riemann–Liouville derivative into its ordinary differential equation. This method is applied to two nonlinear FDEs namely, the time fractional Bogoyavlenskii equation and the space-time fractional Kundu–Eckhaus equation. It is shown that the considered method is very powerful and efficient to solve many nonlinear FDEs.

Lineer Olmayan Kısmi Diferensiyel Denklemlerin exp(-φ(ζ))- Açılım Metodu ile Tam Çözümleri

The aim of this paper is obtaining the exact solutions of the Bogoyavlenskii equation and the modified KdV-Zakharov-Kuznetsev equation with the help of the exp(-φ(ζ))-expansion method. Solutions are obtained with different forms of functions as hyperbolic, trigonometric and rational functions. Discussed method is useful for obtaining solutions of nonlinear equations in mathematical physics and engineering.

Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti-Leon-Pempinelli equations system and Time-fractional Cahn-Allen equation

In this article and for the first time, we introduce and describe Khater method which is a new technique for solving nonlinear partial differential equations (PDEs.). We apply this method for each of the following models Bogoyavlenskii equation, couple Boiti-Leon-Pempinelli system and Time-fractional Cahn-Allen equation. Khater method is very powerful, Effective, felicitous and fabulous method to get exact and solitary wave solution of (PDEs.). Not only just like that but it considers too one of the general methods for solving that kind of equations since it involves some methods as we will see in our discuss of the results. We make a comparison between the results of this new method and another method.

Modified method of simplest equation and its applications to the Bogoyavlenskii equation

In this paper, we construct the exact traveling wave solutions of the Bogoyavlenskii equation using modified method of simplest equation. The simplest equation herewith is a special Riccati equation. In the meantime, the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields.

An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator–prey system

In this article, we apply the exp(-Φ(ξ))-expansion method to construct many families of exact solutions of nonlinear evolution equations (NLEEs) via the nonlinear diffusive predator–prey system and the Bogoyavlenskii equations. These equations can be transformed to nonlinear ordinary differential equations. As a result, some new exact solutions are obtained through the hyperbolic function, the trigonometric function, the exponential functions and the rational forms. If the parameters take specific values, then the solitary waves are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special diffusive predator–prey system and the Bogoyavlenskii equations by the help of programming language Maple.

Exact traveling wave solutions for nonlinear PDEs in mathematical physics using the generalized Kudryashov method

The generalized Kudryashov method is applied in this article for finding the exact solutions of nonlinear partial differential equations (PDEs) in mathematical physics. Solitons and other solutions are given. To illustrate the validity of this method, we apply it to three nonlinear PDEs, namely, the diffusive predator-prey system, the nonlinear Bogoyavlenskii equations and the nonlinear telegraph equation. These equations are related to signal analysis for transmission and propagation of electrical signals. As a result, many analytical exact solutions of these equations are obtained including symmetrical Fibonacci function solutions and hyperbolic function solutions. Physical explanations for some solutions of the given three nonlinear PDEs are obtained. Comparison our new results with the well-known results are given.

Modified extended tanh-function method and its applications to the Bogoyavlenskii equation

In this study, we consider exact traveling wave solutions of the Bogoyavlenskii equation using a modified extended tanh-function method. This method has a broad applicability to many other nonlinear evolution equations in mathematical physics.

English

The modified simple equation method (The modified SEM) is employed to find the exact traveling wave solutions involving parameters for two nonlinear evolution equations, namely, the diffusive predator-prey system and the Bogoyavlenskii equations. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified SEM provides an effective and a more powerful mathematical tool for solving nonlinear partial differential equations in mathematical physics. Our results in this paper are new and different from those obtained in the literature. Key words: A diffusive predator-prey system, the Bogoyavlenskii equation, the modified SEM, traveling wave solutions, solitary wave solutions.

Exact solution of the Bogoyavlenskii equation using the improved tanh-coth method

An improved tanh-coth method is used to obtain exact solutions of the Bogoyavlenskii equation. The obtained solutions are given in a more general form than those obtained in previous literature. One of the most important facts in the paper is that we consider the arbitrary constant derived from a rst integration when the equation is ordinary which is a frequent error in many works which use similar computational methods. Mathematics Subject Classication: 35C05

New Exact Solutions of (2 + 1)-Dimensional Bogoyavlenskii Equation by the Sine-Cosine Method

By using the sine-cosine method proposed recently, we give the exact periodic and soliton solutions of the (2 + 1)-dimensional Bogoyavlenskii equation in this paper. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Bogoyavlenskii equation are successfully obtained. The computation for the method appears to be easier and faster by general mathematical software.

Exact solutions of the Bogoyavlenskii equation using the multiple [formula omitted]-expansion method

In this work, exact traveling wave solutions of the Bogoyavlenskii equation are studied by the (G′G)-expansion method and its variants. The solutions obtained are general solutions which are in the form of hyperbolic, trigonometric and rational functions and a variety of special solutions like kink shaped, antikink shaped, bell type soliton solutions etc., can easily be derived from the general results under certain domain. These methods are more effective and simple than other methods and a number of solutions can be obtained at the same time. In order to visualize the underlying dynamics of the obtained solutions, two and three dimensional plots are drawn.

Periodic and solitary wave solutions of coupled nonlinear wave equations using the first integral method

In this paper, the periodic and solitary wave solutions of two selected systems of nonlinear wave equations are obtained using the first integral method. The long–short-wave interaction system and Bogoyavlenskii equations are considered as examples, and this approach can also be applied to other coupled nonlinear differential equations.

On exact solutions of the Bogoyavlenskii equation

Exact solutions for the Bogoyavlenskii equation are studied by the travelling wave method and the singular manifold method. It is found that the linear superposition of the shock wave solution and the complex solitary wave solution for the physical field is still a solution of the equation of interest, except for a phase-shift. The dromion-like structures with elastic and nonelastic interactions are found.