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.     

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.     

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.     

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.     

A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations

A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer–Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations.     

一类广义非线性强阻尼扰动发展方程的行波解

研究了一类非线性强阻尼广义扰动发展方程问题.它们在数学、力学、物理学等领域中广泛出现.首先,引入一个行波变换,把相应的偏微分方程问题转化为行波方程问题并求出原典型问题的精确解.再用小参数方法和引入伸长变量构造了问题的渐近解.最后, 用泛函分析的不动点理论证明了原非线性强阻尼广义扰动发展方程初值问题渐近行波解的存在性,并证明渐近解具有较高的精度和一致有效性.该文求得的渐近解是一个解析展开式, 所以它还可继续进行解析运算, 而单纯用数值模拟的方法是不行的.     

(3+1)-维广义随机KP方程的精确解

利用统一方式构造非线性偏微分方程行波解的广义Jacobi椭圆函数展开法和Hermite变换来研究(3+1)-维广义随机KP方程,给出了它的随机对偶周期和多孤子解.     

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     

一类非线性波动方程的显式精确解

本文用直接方法和假设的一种结合求出了一类较广泛的非线性波动方程ｕｔｔ－ａ１ｕｘｘ＋ａ２ｕｔ＋ａ３ｕ＋ａ４ｕＳ＾２＋ａ５ｕ＾３＝０的一些显式精确行波解,贱个有重要的非线性数学物理方程,如φ＾４方程,Ｋｌｅｉｎ－Ｇｏｒｄｏｎ方程,Ｓｉｎｅ－Ｇｏｒｄｏｎ方程,及Ｓｉｎｈ－Ｇｏｒｄｏｎ方程的近似,Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ－Ｈｉｇｇｓ方程,Ｄｕｆｆｉｎｇ方程,非线性电报方程等都可作为该方程的特殊情形得     

New exact complex travelling wave solutions to nonlinear Schrödinger (NLS) equation

Two improved direct algebraic methods for constructing exact complex travelling wave solutions of nonlinear partial differential equations are presented. These improved methods are applied to NLS equation, and then new types exact complex solutions are obtained.     

A new algebraic procedure to construct exact solutions of nonlinear differential-difference equations

This paper presents a new algebraic procedure to construct exact solutions of selected nonlinear differential-difference equations. The discrete sine-Gordon equation and differential-difference asymmetric Nizhnik-Novikov-Veselov equations are chosen as examples to illustrate the efficiency and effectiveness of the new procedure, where various types of exact travelling wave solutions for these nonlinear differential-difference equations have been constructed. It is anticipated that the new procedure can also be used to produce solutions for other nonlinear differential-difference equations.     

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.     

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.     

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.     

On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation are successfully obtained, which contain soliton-like and periodic-like solutions This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

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.     

Observations on the class of “Balancing Principle” for nonlinear PDEs that can be treated by the auxiliary equation method

There is collection of methods for finding explicit travelling wave solutions of nonlinear partial differential equations (NLPDEs) in the literature. Quite a large amount of these methods employ “Balancing Principle” in their methodology and they work for positive integer values only. Yet there is no well established formula for balancing and there is no specific rule to determine adequate balancing principle that works for any number which is positive/or negative and/or rational. In this study, we propose a new balancing principle which works for large range of the numbers either positive/or negative and/or rational that makes it possible to obtain new and original explicit travelling wave solutions of nonlinear partial differential equations (NLPDEs).     

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.     

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.     

An observation on the periodic solutions to nonlinear physical models by means of the auxiliary equation with a sixth-degree nonlinear term

It is a fact that in auxiliary equation methods, the exact solutions of different types of auxiliary equations may produce new types of exact travelling wave solutions to nonlinear equations. In this manner, various auxiliary equations of first-order nonlinear ordinary differential equation with distinct-degree nonlinear terms are examined and, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with sixth-degree nonlinear term are presented. Consequently, the novel exact solutions of the generalized Klein–Gordon equation and the active-dissipative dispersive media equation are found out for illustration purposes. They are also applicable, where conventional perturbation method fails to provide any solution of the nonlinear problems under study.