Travelling wave solutions of nonlinear partial equations by using the first integral method

In this paper, the first integral method is employed for constructing the new exact travelling wave solutions of nonlinear partial differential equations. The power of this manageable method is confirmed by applying it for two selected nonlinear partial equations. This approach can also be applied to other systems of nonlinear differential equations.     

任意方程法及BKP方程新的精确解

In this paper,auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equation with the aid of symbolic computation.We study the(2+1)-dimensional BKP equation and get a series of new types of traveling wave solutions.The method used here can be also extended to other nonlinear partial differential equations.     

Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation

In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Bäcklund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations.     

A Note on Some Sub-Equation Methods and New Types of Exact Travelling Wave Solutions for Two Nonlinear Partial Differential Equations

Sub-equation methods are used for constructing exact travelling wave solutions of nonlinear partial differential equations. The key idea of these methods is to take full advantage of all kinds of special solutions of sub-equation, which is usually a nonlinear ordinary differential equation. We present a function transformation which not only gives us a clear relation among these sub-equation methods, but also can be used to obtain the general solutions of these sub-equations. And then new exact travelling wave solutions of the CKdV-MKdV equation and the CKdV equations as applications of this transformation are obtained, and the approach presented in this paper can be also applied to other nonlinear partial differential equations.      

Construction of a series of travelling wave solutions to nonlinear equations

In this paper, based on new auxiliary ordinary differential equation with a sixth-degree nonlinear term, we study the (1 + 1)-dimensional combined KdV–MKdV equation, Hirota equation and (2 + 1)-dimensional Davey–Stewartson equation. Then, a series of new types of travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.     

Exact travelling wave solutions of the discrete sine–Gordon equation obtained via the exp-function method

In this paper, we generalize the exp-function method, which was used to find new exact travelling wave solutions of nonlinear partial differential equations (NPDEs) or coupled nonlinear partial differential equations, to nonlinear differential–difference equations (NDDEs). As an illustration, two series of exact travelling wave solutions of the discrete sine–Gordon equation are obtained by means of the exp-function method. As some special examples, these new exact travelling wave solutions can degenerate into the kink-type solitary wave solutions reported in the open literature.     

基于Riccati-Bernoulli辅助常微分方程的Davey-Stewartson方程的行波解

Riccati-Bernoulli辅助常微分方程方法可以用来构造非线性偏微分方程的行波解.利用行波变换,将非线性偏微分方程化为非线性常微分方程, 再利用Riccati-Bernoulli方程将非线性常微分方程化为非线性代数方程组, 求解非线性代数方程组就能直接得到非线性偏微分方程的行波解.对Davey-Stewartson方程应用这种方法, 得到了该方程的精确行波解.同时也得到了该方程的一个Backlund变换.所得结果与首次积分法的结果作了比较.Riccati-Bernoulli辅助常微分方程方法是一种简单、有效地求解非线性偏微分方程精确解的方法.     

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.     

Auxiliary equation method for solving nonlinear Wick-type partial differential equations

Using the solutions of an auxiliary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some Wick-type nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained. In addition, the links between Wick-type partial differential equations and variable coefficient partial differential equations are also clarified generally.     

非线性系统的精确解

Based on the computerized symbolic, a new generalized tanh functions method is used for constructing exact travelling wave solutions of nonlinear partial differential equations （PDES） in a unified way. The main idea of our method is to take full advantage of an auxiliary ordinary differential equation which has more new solutions. At the same time, we present a more general transformation, which is a generalized method for finding more types of travelling wave solutions of nonlinear evolution equations （NLEEs）. More new exact travelling wave solutions to two nonlinear systems are explicitly obtained.     

New exact solutions to MKDV-Burgers equation and (2 + 1)-dimensional dispersive long wave equation via extended Riccati equation method

In this paper, with the aid of symbolic computation and a general ansätz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansätz. The method can also be applied to other nonlinear partial differential equations.     

A note on the elliptic equation method

The elliptic equation method is improved for constructing exact travelling wave solutions of nonlinear partial differential equations (PDEs). The rational forms of Jacobi elliptic functions are presented. By using new Jacobi elliptic function solutions of the elliptic equation, new doubly periodic solutions are obtained for some important PDEs. This method can be applied to many other nonlinear PDEs.     

A new auxiliary function method for solving the generalized coupled Hirota–Satsuma KdV system

In this letter, a new auxiliary function method is presented for constructing exact travelling wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the elliptic equation to construct exact travelling wave solutions of nonlinear partial differential equations. More new exact travelling wave solutions are obtained for the generalized coupled Hirota–Satsuma KdV system.     

Application of Exp-function method to wave solutions of the Sine-Gordon and Ostrovsky equations

In this work, the Exp-function method is employed to find new wave solutions for the Sine-Gordon and Ostrovsky equation. The equations are simplified to the nonlinear partial differential equations and then different types of exact solutions are extracted by this method. It is shown that the Exp-function method is a powerful analytical method for solving other nonlinear equations occurring in nonlinear physical phenomena. Results are presented in contour plots that show the different values of effective parameters on the velocity profiles.     

More solutions of the auxiliary equation to get the solutions for a class of nonlinear partial differential equations

In this paper, a new auxiliary function method is presented for constructing exact travelling wave solutions of nonlinear evolution equations. By the relationships of Jacobi elliptic functions, we get more solutions of the auxiliary equation compared with El-Wakila and Abdou (2006) [22]. So, more new exact travelling wave solutions are obtained for a class of nonlinear partial differential equations.     

Extended tanh-function Metho d for Solving Traveling Wave Solutions of Nonlinear Kundu Equation

In this paper, by using the balancing method and the extended tanh-function method, we obtain the exact traveling wave solutions of Kundu equation with fifth-order nonlinear term. Applications of this method to some other nonlinear partial differential equations are also presented.     

累次齐次平衡法及其应用

在求非线性偏微分方程精确解的过程中两次使用了齐次平衡法(称为累次齐次平衡法),解决了齐次平衡法求解少的不足,从而改进了齐次平衡法.以高阶(2+1)维Kadomtsev-Petviashvili方程和变异的Boussinesq方程为应用实例,说明使用累次齐次平衡法可以求得大量的精确解,其中许多解是新解或覆盖了其他方法所得的解.方法可应用于大量的非线性物理模型.     

一类Burgers方程的孤立波解

Burgers方程在工程上有着重要的应用,它可以用来描述湍流、车队的交通流、氏族的随机迁移、化学工程中的分离等现象,对Burgers方程求解方法的研究有着重要的现实意义.对Burgers方程求解主要是应用差分和微分两方面的方法来展开求解的,1/G展开法是近年来发展起来的求解非线性偏微分方程的一种较为有效的微分解法.采用微分方程方面的方法,利用1/G展开法对一类Burgers方程进行求解,得到了此方程的一类孤立波解和扭曲波解,同时描绘出解的图像并分析解的结构和变化趋势.     

非线性耦合Klein-Gordon方程组和耦合Schr\"{o}dinger-Boussinesq方程组的双行波解

本文提出了一种全新复合$(\frac{G''}{G})$展开方法,运用这种新方法并借助符号计算软件构造了非线性耦合Klein-Gordon方程组和耦合Schr\"{o}dinger-Boussinesq方程组的多种双行波解,包括双双曲正切函数解,双正切函数解,双有理函数解以及它们的混合解. 复合$(\frac{G''}{G})$展开方法不但直接有效地求出了两类非线性偏微分方程的双行波解,而且扩大了解的范围.这种新方法对于研究非线性偏微分方程具有广泛的应用意义.     

Reduced differential transform method for nonlinear integral member of Kadomtsev–Petviashvili hierarchy differential equations

The main objective of this paper is to use the reduced differential to transform method (RDTM) for finding the analytical approximate solutions of two integral members of nonlinear Kadomtsev–Petviashvili (KP) hierarchy equations. Comparing the approximate solutions which obtained by RDTM with the exact solutions to show that the RDTM is quite accurate, reliable and can be applied for many other nonlinear partial differential equations. The RDTM produces a solution with few and easy computation. This method is a simple and efficient method for solving the nonlinear partial differential equations. The analysis shows that our analytical approximate solutions converge very rapidly to the exact solutions.