Exact solutions of nonlinear evolution equations of the AKNS class

1989 ◽  
Vol 30 (3) ◽  
pp. 313-325 ◽  
Author(s):  
W. L. Chan ◽  
Yu-Kun Zheng
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Mkdv Equation ◽  
Nonlinear Evolution Equations ◽  
Sine Gordon Equation ◽  
First Order ◽  
Linear Eigenvalue Problem ◽  
The Kdv Equation ◽  
Akns System

AbstractThe problem of obtaining explicit and exact solutions of soliton equations of the AKNS class is considered. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system; that is, a linear eigenvalue problem in the form of a system of first order partial differential equations. The method of characteristics is used and Bäcklund transformations are employed to generate new solutions from the old. Thus, families of new solutions for the KdV equation, the mKdV equation, the sine-Gordon equation and the nonlinear Schrôdinger equation are obtained, avoiding the solution of some Riccati equations. Our results in the KdV case include those obtained recently by other investigators.

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 Exact Solutions for the KdV Equation with Forcing Term by the Generalized tanh-coth Method and the (G′/G)-Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Jalil Manafian ◽  
Monireh Nasrollahpour
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Problems ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Forcing Term ◽  
The Kdv Equation

An application of the generalized tanh-coth method and the (G′/G)-expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. These methods are used for the KdV equation with forcing term. The generalized tanh-coth method and the (G′/G)-expansion method were used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. It is shown that the generalized tanh-coth method and the (G′/G)-expansion method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear problems.

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 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.

scholarly journals Some exact solutions to the generalized Korteweg–de Vries equation and the system of shallow water wave equations

2013 ◽  
Vol 18 (1) ◽  
pp. 27-36 ◽  
Author(s):  
Ihsan Timuçin Dolapci ◽  
Ahmet Yıldırım
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Nonlinear Differential Equations ◽  
Nonlinear Evolution Equations ◽  
Transform Method ◽  
Shallow Water Wave ◽  
Solitary Solutions

In this paper, we establish exact solutions for nonlinear evolution equations in mathematical physics. The exp-transform method is proposed to seek solitary solutions, periodic solutions and compaction-like solutions of nonlinear differential equations. The generalized KdV equation and the system of the shallow water wave equation are chosen to illustrate the effectiveness and convenience of the method.

Abundant New Exact Solutions of the Coupled Potential KdV Equation and the Modified KdV-Type Equation

2001 ◽  
Vol 56 (12) ◽  
pp. 809-815
Author(s):  
Zhenya Yan
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Type Equation ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Rational Solutions ◽  
Soliton Theory ◽  
Kdv Type Equations

Abstract Exact solutions of nonlinear evolution equations (NLEEs)in soliton theory and their applications are studied. A powerful method is established to search for exact travelling wave solutions of NLEEs. We chose the coupled potential KdV equation and modified KdV-type equations presented by Foursov to illustrate the approach with the aid of Maple. As a result, eight families of exact solutions of the coupled potential KdV equation and nine families of exact solutions of the modified KdV-type equations are obtained, which contain new kink-like soliton solutions, kink­ shaped solitons, bell-shaped solitons, periodic solutions, rational solutions and singular solitons. The properties of the solutions are shown in figures.

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.

Sign in / Sign up

Export Citation Format

Share Document