（G′/G）-Expansion Method Equivalent to Extended Tanh Function Method

In a recent article [Physics Letters A 372 （2008） 417], Wang et al. proposed a method, which is called the （G′/G）-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota-Satsuma equations are obtained by using this method. They think the （G′/G）-expansion method is a new method and more travelling wave solutions of many nonlinear evolution equations can be obtained. In this paper, we will show that the （G′/G）-expansion method is equivalent to the extended tanh function method.     

Lie group analysis,numerical and non-traveling wave solutions for the(2+1)-dimensional diffusion–advection equation with variable coefficients

In this paper, the variable-coefficient diffusion–advection(DA) equation, which arises in modeling various physical phenomena, is studied by the Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation, and then exact and numerical solutions are reported for the reduced second-order nonlinear ordinary differential equations. Further, an extended(G /G)-expansion method is applied to the DA equation to construct some new non-traveling wave solutions.     

A novel （G＇/G）-expansion method and its application to the Boussinesq equation

In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new （G＇/G）-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.     

A study on stochastic longitudinal wave equation in a magneto-electro-elastic annular bar to find the analytical solutions

In this paper, we set up dynamic solitary perturb solutions of a unidirectional stochastic longitudinal wave equation in a magneto-electro-elastic annular bar by a feasible, useful, and influential method named the dual(G’/G, 1/G)-expansion method. Computer software, like Mathematica, is used to complete this discussion. The obtained solutions of the proposed equation are classified into trigonometric, hyperbolic, and rational types which play an important role in searching for numerous scientif...     

Exact Solutions to （2＋1）-Dimensional Kaup-Kupershmidt Equation

In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the （2＋1）-dimensional Kaup-Kupershmidt （KK） equation are presented. We successfully obtain more general exact travelling wave solutions for （2＋ 1）-dimensional KK equation by the symmetry method and the （G1/G）-expansion method. Consequently, we find some new solutions of （2＋1）-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions.     

Explicit solutions of nonlinear wave equation systems

We apply the (G’/G)-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. We highlight the power of the (G’/G)-expansion method in providing generalized solitary wave solutions of different physical structures. It is shown that the (G’/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.     

Lie group analysis,numerical and non-traveling wave solutions for the （2＋l）-dimensional diffusion-advection equation with variable coefficients

In this paper, the variable-coefficient diffusion-advection （DA） equation, which arises in modeling various physical phenomena, is studied by the Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation, and then exact and numerical solutions are reported for the reduced second-order nonlinear ordinary differential equations. Further, an extended （Gl/G）-expansion method is applied to the DA equation to construct some new non-traveling wave solutions.     

(G′/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics

In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.     

Singular solitons and other solutions to a couple of nonlinear wave equations

This paper addresses the extended （G′/G）-expansion method and applies it to a couple of nonlinear wave equations. These equations are modified the Benjamin-Bona-Mahoney equation and the Boussinesq equation. This extended method reveals several solutions to these equations. Additionally, the singular soliton solutions are revealed, for these two equations, with the aid of the ansatz method.     

Generalized and Improved （G＇/G）-Expansion Method Combined with Jacobi Elliptic Equation

In this article, we propose an alternative approach of the generalized and improved （G＇/G）-expansion method and build some new exact traveling wave solutions of three nonlinear evolution equations, namely the Boiti- Leon-Pempinelle equation, the Pochhammer-Chree equations and the Painleve integrable Burgers equation with free parameters. When the free parameters receive particular values, solitary wave solutions are constructed from the traveling waves. We use the Jacob/elliptic equation as an auxiliary equation in place of the second order linear equation. It is established that the proposed algorithm offers a further influential mathematical tool for constructing exact solutions of nonlinear evolution equations.     

A connection between the(G'/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation

Recently the (G'/G)-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the (G'/G)-expansion method is a special form of the truncated Painlevé expansion method by introducing an intermediate expansion method. Then the generalized (G'/G)--(G'/G) expansion method is naturally derived from the standpoint of the nonstandard truncated Painlevé expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the (G'/G)-expansion method.     

A New Algebraic Method and Its Application to Nonlinear Klein-Gordon Equation

In terms of the solutions of the generalized Riccati equation, a new algebraic method, which contains the terms of radical expression of functions f（ξ）, is constructed to explore the new exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to nonlinear Klein-Gordon equation, and some new exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.     

Exact Solutions of （2＋1）-Dimensional Boiti-Leon-Pempinelle Equation with （G′/G）-Expansion Method

In this paper, we construct exact solutions for the （2＋1）-dimensional Boiti-Leon-Pempinelle equation by using the （G′/G）-expansion method, and with the help of Maple. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. This method can be applied to other higher-dimensional nonlinear partial differential equations.     

Exact solutions for nonlinear partial fractional differential equations

In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.     

A Generalized Algebraic Method for Constructing a Series of Explicit Exact Solutions of a （1＋1）-Dimensional Dispersive Long Wave Equation

Making use of a new and more general ansatz, we present the generalized algebraic method to uniformly construct a series of new and general travelling wave solution for nonlinear partial differential equations. As an application of the method, we choose a (1 1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by the method proposed by Fan [E. Fan, Comput. Phys. Commun. 153 (2003) 17] and find other new and more general solutions at the same time, which include polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solutions, hyperbolic and soliton solutions, Jacobi and Weierstrass doubly periodic wave solutions.     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

Abstract In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f （ξ）, is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

New doubly periodic and multiple soliton solutions of the generalized （3＋l）-dimensional KP equation with variable coefficients

A new generalized Jacobi elliptic function method is used to construct the exact travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new doubly periodic and multiple soliton solutions are obtained for the generalized (3+1)-dimensional Kronig－Penny (KP) equation with variable coefficients. This method can be applied to other equations with variable coefficients.     

New Multiple Soliton-like Solutions to （3＋1）-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1)-dimensional Burgers equation with variable coefficients.     

NEW EXPLICIT AND EXACT TRAVELLING WAVE SOLUTIONS FOR A COMPOUND KdV－BURGERS EQUATION

In this paper, new explicit and exact travelling wave solutions for a compound KdV－Burgers equation are obtained by using the hyperbola function method and the Wu elimination method, which include new solitary wave solutions and periodic solutions. Particularly important cases of the equation, such as the compound KdV, mKdV－Burgers and mKdV equations can be solved by this method. The method can also solve other nonlinear partial differential equations.     

A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model

Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony （mBBM） model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations.