Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method

The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov （KPP） equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

A series of new exact solutions of nonlinear evolution equations

With the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (2 + 1)-dimensional Konopelchenko-Dubrovsky equations

In this paper, new exact solutions with two arbitrary functions of the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations are obtained by means of the Riccati equation and its generalized solitary wave solutions constructed by the Exp-function method. It is shown that the Exp-function method provides us with a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics.     

New exact travelling wave solutions using modified extended tanh-function method

Based on computerized symbolic computation and modified extended tanh-function method for constructing a new exact travelling wave solutions of nonlinear evolution equations (NEEs) is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some (NEEs) with mathematical physics interests. As a results, we can successfully recover the previously known solitary wave solutions that had been found by the tanh-function method and other more sophisticated methods.     

New Explicit and Exact Solutions for the Klein-Gordon-Zakharov Equations

In this paper, based on the generalized Jacobi elliptic function expansion method, we obtain abundant new explicit and exact solutions of the Klein-Gordon- Zakharov equations, which degenerate to solitary wave solutions and triangle function solutions in the limit cases, showing that this new method is more powerful to seek exact solutions of nonlinear partial differential equations in mathematical physics.     

Explicit solutions of (2 + 1)-dimensional nonlinear KP-BBM equation by using Exp-function method

This paper is devoted to studying the (2 + 1)-dimensional KP-BBM wave equation. Exp-function method is used to conduct the analysis. The generalized solitary solutions, periodic solutions and other exact solutions for the (2 + 1)-dimensional KP-BBM wave equation are obtained via this method with the aid of symbolic computational system. It is also shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

The Riccati Equation Method Combined with the Generalized Extended $(G''/G)$-Expansion Method for Solving the Nonlinear KPP Equation

An analytic study of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation is presented in this paper. The Riccati equation method combined with the generalized extended $(G''/G)$-expansion method is an interesting approach to find more general exact solutions of the nonlinear evolution equations in mathematical physics. We obtain the traveling wave solutions involving parameters, which are expressed by the hyperbolic and trigonometric function solutions. When the parameters are taken as special values, the solitary and periodic wave solutions are given. Comparison of our new results in this paper with the well-known results are given.     

New periodic wave solutions for nonlinear evolution equations with variable coefficients via mapping method

An extended mapping method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for nonlinear evolution equations arising in physics, namely, generalized Zakharov Kuznetsov equation with variable coefficients. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations with variable coefficients arising in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

New periodic wave solutions via extended mapping method

In this paper, an extended mapping method with a computerized symbolic computation is used for constructing new periodic wave solutions for two nonlinear evolution equations arising in mathematical physics, namely, generalized nonlinear Schroedinger equation and generalized-Zakharov equations. As a result, many exact travelling wave solutions are obtained which include new periodic wave solutions, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also applied to other nonlinear evolution equations.     

The extended Riccati equation method for travelling wave solutions of ZK equation

In this article, the extended Riccati equation method is applied to seeking more general exact travelling wave solutions of the ZK equation. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters are taken as special values, the solitary wave solutions are obtained from the hyperbolic function solutions. Similarly, the periodic wave solutions are also obtained from the trigonometric function solutions. The approach developed in this paper is effective and it may also be used for solving many other nonlinear evolution equations in mathematical physics.     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

Application of the improved F-expansion method with Riccati equation to find the exact solution of the nonlinear evolution equations

In this article, we pay attention to the analytical method named, the improved F-expansion method combined with Riccati equation for finding the exact traveling wave solutions of the Benney–Luke equation and the Phi-4 equation. By means of this method we have explored three classes of explicit solutions-hyperbolic, trigonometric and rational solutions with some free parameters. When the parameters are taken as special values, the solitary wave solutions are originated from the traveling wave solutions. Our outcomes disclose that this method is very active and forthright way of formulating the exact solutions of nonlinear evolution equations arising in mathematical physics and engineering.     

Explicit and exact travelling wave solutions for the generalized derivative Schrödinger equation

In this paper, a new auxiliary equation expansion method and its algorithm is proposed by studying a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. Being concise and straightforward, the method is applied to the generalized derivative Schrödinger equation. As a result, some new exact travelling wave solutions are obtained which include bright and dark solitary wave solutions, triangular periodic wave solutions and singular solutions. This algorithm can also be applied to other nonlinear wave equations in mathematical physics.     

The extended Jacobian elliptic function expansion method and its application in the generalized Hirota–Satsuma coupled KdV system

In this paper an extended Jacobian elliptic function expansion method, which is a direct and more powerful method, is used to construct more new exact doubly periodic solutions of the generalized Hirota–Satsuma coupled KdV system by using symbolic computation. As a result, sixteen families of new doubly periodic solutions are obtained which shows that the method is more powerful. When the modulus of the Jacobian elliptic functions m→1 or 0, the corresponding six solitary wave solutions and six trigonometric function (singly periodic) solutions are also found. The method is also applied to other higher-dimensional nonlinear evolution equations in mathematical physics.     

Soliton and Periodic Wave Solutions of the Nonlinear Loaded (3+1)-Dimensional Version of the Benjamin-Ono Equation by Functional Variable Method

In this article, we establish new travelling wave solutions for the nonlinear loaded (3+1)-dimensional version of the Benjamin-Ono equation by the functional variable method. The performance of this method is reliable and effective and the method provides the exact solitary wave solutions and periodic wave solutions. The solution procedure is very simple and the traveling wave solutions are expressed by hyperbolic functions and trigonometric functions. After visualizing the graphs of the soliton solutions and the periodic wave solutions, the use of distinct values of random parameters is demonstrated to better understand their physical features. It has been shown that the method provides a very effective and powerful mathematical tool for solving nonlinear equations in mathematical physics.     

A new extended Riccati equation rational expansion method and its application

In this paper, a new extended Riccati equation rational expansion method is suggested to constructing multiple exact solutions for nonlinear evolution equations. The validity and reliability of the method is tested by its application to the dispersive long wave system and the Broer–Kaup–Kupershmidt system. The method can be applied to other nonlinear evolution equations in mathematical physics.     

A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order

In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations.     

Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method

In this paper, we establish exact solutions for (2 + 1)-dimensional nonlinear evolution equations. The sine-cosine method is used to construct exact periodic and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Boussinesq, breaking soliton and BKP equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.     

Modified extended tanh-function method for solving nonlinear partial differential equations

Based on computerized symbolic computation, modified extended tanh-method for constructing multiple travelling wave solutions of nonlinear evolution equations is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some nonlinear evolution equations in mathematical physics such as the nonlinear partial differential equation, nonlinear Fisher-type equation, ZK-BBM equation, generalized Burgers–Fisher equation and Drinfeld–Sokolov system. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods.     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.