scholarly journals Further Results about Traveling Wave Exact Solutions of the (2+1)-Dimensional Modified KdV Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yang Yang ◽  
Jian-ming Qi ◽  
Xue-hua Tang ◽  
Yong-yi Gu
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Mkdv Equation ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
Modified Kdv Equation

We used the complex method and the exp(-ϕ(z))-expansion method to find exact solutions of the (2+1)-dimensional mKdV equation. Through the maple software, we acquire some exact solutions. We have faith in that this method exhibited in this paper can be used to solve some nonlinear evolution equations in mathematical physics. Finally, we show some simulated pictures plotted by the maple software to illustrate our results.

scholarly journals New Jacobi Elliptic Function Solutions for the Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Yun-Mei Zhao
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function

A generalized(G′/G)-expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics.

scholarly journals Some Methods about Finding the Exact Solutions of Nonlinear Modified BBM Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Fan Niu ◽  
Jianming Qi ◽  
Zhiyong Zhou
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
First Time ◽  
Bbm Equation

Finding exact solutions of nonlinear equations plays an important role in nonlinear science, especially in engineering and mathematical physics. In this paper, we employed the complex method to get eight exact solutions of the modified BBM equation for the first time, including two elliptic function solutions, two simply periodic solutions, and four rational function solutions. We used the exp − ϕ z -expansion methods to get fourteen forms of solutions of the modified BBM equation. We also used the sine-cosine method to obtain eight styles’ exact solutions of the modified BBM equation. Only the complex method can obtain elliptic function solutions. We believe that the complex method presented in this paper can be more effectively applied to seek solutions of other nonlinear evolution equations.

scholarly journals Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fu Zhang ◽  
Jian-ming Qi ◽  
Wen-jun Yuan
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
Wave Solutions ◽  
Periodic Traveling Wave

We employ the complex method to obtain all meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutionsvs,3(z)which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

AN EFFICIENT METHOD FOR FINDING THE EXACT SOLUTION OF NONLINEAR EVOLUTION EQUATIONS

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1703-1706 ◽  
Author(s):  
XIQIANG ZHAO ◽  
DENGBIN TANG ◽  
CHANG SHU
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

In this paper, based on the idea of the homogeneous balance method, the special truncated expansion method is improved. The Burgers-KdV equation is discussed and its many exact solutions are obtained with the computerized symbolic computation system Mathematica. Our method can be applied to finding exact solutions for other nonlinear partial differential equations too.

scholarly journals The Two-Variable -Expansion Method for Solving the Nonlinear KdV-mKdV Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
E. M. E. Zayed ◽  
M. A. M. Abdelaziz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Mkdv Equation ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
Powerful Mathematical Tool

We apply the two-variable (, )-expansion method to construct new exact traveling wave solutions with parameters of the nonlinear ()-dimensional KdV-mKdV equation. This method can be thought of as the generalization of the well-known ()-expansion method given recently by M. Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions of this equation are rediscovered from the traveling waves. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

scholarly journals F-Expansion Method and New Exact Solutions of the Schrödinger-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ali Filiz ◽  
Mehmet Ekici ◽  
Abdullah Sonmezoglu
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function ◽  
Weierstrass Elliptic Function ◽  
New Exact Solutions

F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the Schrödinger-KdV equation with a source to illustrate the validity and advantages of the proposed method. A number of Jacobi-elliptic function solutions are obtained including the Weierstrass-elliptic function solutions. When the modulusmof Jacobi-elliptic function approaches to 1 and 0, soliton-like solutions and trigonometric-function solutions are also obtained, respectively. The proposed method is a straightforward, short, promising, and powerful method for the nonlinear evolution equations in mathematical physics.

Extended Jacobian Elliptic Function Expansion Method and Its Applications in Mathematical Physics

2016 ◽  
pp. 3585-3592
Author(s):  
Mostafa Khater
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Validity And Reliability ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to some nonlinear evolution equations which play an important role inmathematical physics.

Geometrical Properties and Exact Solutions of Three (3+1)-Dimensional Nonlinear Evolution Equations in Mathematical Physics Using Different Expansion Methods

2019 ◽  
pp. 1-19
Author(s):  
A. R. Shehata ◽  
Safaa S. M. Abu-Amra
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Shallow Water Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions

In this article, A Variation of -Expansion Method and -Expansion Method have been applied to find the traveling wave solutions of the (3+1)-dimensional Zakhrov-Kuznetsov (ZK) equation, the (3+1)-dimensional Potential-YTSF Equation and the (3+1)-dimensional generalized Shallow water equation. The efficiency of these methods for finding the exact solutions have been demonstrated. As a result, some new exact traveling wave solutions are obtained which include solitary wave solutions. It is shown that the methods are effective and can be used for many other Nonlinear Evolution Equations (NLEEs) in mathematical physics. In this article, A Variation of -Expansion Method and -Expansion Method have been applied to find the traveling wave solutions of the (3+1)-dimensional Zakhrov-Kuznetsov (ZK) equation, the (3+1)-dimensional Potential-YTSF Equation and the (3+1)-dimensional generalized Shallow water equation. The efficiency of these methods for finding the exact solutions have been demonstrated. As a result, some new exact traveling wave solutions are obtained which include solitary wave solutions. It is shown that the methods are effective and can be used for many other Nonlinear Evolution Equations (NLEEs) in mathematical physics.

scholarly journals New Jacobi Elliptic Function Solutions for the Kudryashov-Sinelshchikov Equation Using Improved F-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yinghui He
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

Based on the F-expansion method with a new subequation, an improved F-expansion method is introduced. As illustrative examples, some new exact solutions expressed by the Jacobi elliptic function of the Kudryashov-Sinelshchikov equation are obtained. When the modulusmof the Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function and trigonometric function can also be obtained. The method is straightforward and concise and is promising and powerful for other nonlinear evolution equations in mathematical physics.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

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