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.     

Explicit exact solutions for (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, we construct explicit exact solutions for the coupled Boiti–Leon–Pempinelli equation (BLP equation) by using a extended tanh method and symbolic computation system Mathematica. By means of the method, many new exact travelling wave solutions for the BLP system are successfully obtained. the extended tanh method can be applied to other higher-dimensional coupled nonlinear evolution equations in mathematical physics.     

The extended homogeneous balance method and its applications for a class of nonlinear evolution equations

The extended homogeneous balance method with the aid of computer algebraic system Maple, is proposed for seeking the travelling wave solutions for a class of nonlinear evolution equations, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear evolution equations, respectively. Many new exact travelling wave solutions are successfully obtained. The method is straightforward and concise, and it can be also applied to other nonlinear evolution 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.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution equations.     

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.     

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.     

Generalized Extended Tanh-Function Method for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher's equation, the nonlinear schrödinger equation to illustrate the validity and advantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.     

Symbolic computation and construction of soliton-like solutions for a breaking soliton equation

Based on the symbolic computation system––Maple and a Riccati equation, by introducing a new more general ansätz than the ansätz in the tanh method, extended tanh-function method, modified extended tanh-function method, generalized tanh method and generalized hyperbolic-function method, we propose a generalized Riccati equation expansion method for searching for exact soliton-like solutions of nonlinear evolution equations and implemented in computer symbolic system––Maple. Making use of our method, we study a typical breaking soliton equation and obtain new families of exact solutions, which include the nontravelling wave’ and coefficient function’ soliton-like solutions, singular soliton-like solutions and periodic solutions. The arbitrary functions of some solutions are taken to be some special constants or functions, the known solutions of this equation can be recovered.     

Applications of the extended tanh method for coupled nonlinear evolution equations

In this work, we establish exact solutions for coupled nonlinear evolution equations. The extended tanh method is used to construct solitary and soliton solutions of nonlinear evolution equations. The extended tanh method presents a wider applicability for handling nonlinear wave equations.     

On traveling wave solutions to combined KdV–mKdV equation and modified Burgers–KdV equation

In this work, we established exact solutions for some nonlinear evolution equations. The extended tanh method was used to construct solitary and soliton solutions of nonlinear evolution equations. The extended tanh method presents a wider applicability for handling nonlinear wave equations.     

New exact solutions for some power law nonlinear diffusion equation

An extended auxiliary equation method for exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. This, together with a convenient characterization, affords new exact traveling wave solutions of some classes of nonlinear power law diffusion equations to be obtained.     

Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the new Hamiltonian amplitude equation introduced by Wadati et al. When the modulus m approaches to 1 and 0, then the hyperbolic function solutions (including the solitary wave solutions) and trigonometric function solutions are also given respectively. As the parameter ε goes to zero, the new Hamiltonian amplitude equation becomes the well-known nonlinear Schrödinger equation (NLS), and at least there are 37 kinds of solutions of NLS can be derived from the solutions of the new Hamiltonian amplitude equation.     

Travelling wave solutions of some classes of nonlinear evolution equations in (1+1) and (2+1) dimensions

The tanh method is proposed to find travelling wave solutions in (1+1) and (2+1) dimensional wave equations. It can be extended to solve a whole family of modified Korteweg–de Vries type of equations, higher dimensional wave equations and nonlinear evolution equations.     

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     

双函数法及一类非线性发展方程的精确行波解

给出一种求解非线性发展方程精确行波解的新方法：双函数法。使用此方法，获得了一类非线性发展方程的许多精确行波解，其中包括孤波解和周期解，推广了文献用其它方法取得的结果，同时还获得了许多新的弧波解和周期解，借助于Mathemat-ica，此方法能部分地在计算机上实现。     

Polygons of differential equations for finding exact solutions

A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of polygons corresponding to nonlinear differential equations. It allows one to express exact solutions of the equation studied through solutions of another equation using properties of the basic equation itself. The ideas of power geometry are used and developed. Our approach has a pictorial interpretation, which is illustrative and effective. The method can be also applied for finding transformations between solutions of differential equations. To demonstrate the method application exact solutions of several equations are found. These equations are: the Korteveg–de Vries–Burgers equation, the generalized Kuramoto–Sivashinsky equation, the fourth-order nonlinear evolution equation, the fifth-order Korteveg–de Vries equation, the fifth-order modified Korteveg–de Vries equation and the sixth-order nonlinear evolution equation describing turbulent processes. Some new exact solutions of nonlinear evolution equations are given.     

On the nontrivial non-travelling wave profiles of nonlinear evolution and wave equations

The overall aim of the present paper is to find and analyze the new non-travelling wave solutions of the nonlinear evolution and wave equations. With the aid of symbolic computation and based on the generalized extended tanh-function method, we propose the newly extended tanh-function expansion algorithm and get many new non-travelling wave solutions of the (2 + 1)-dimensional Broer–Kaup–Kupershmidt equations. The solutions which we obtain are more abundant than the solutions which the generalized extended tanh-function method gets. At the same time, the solutions contain arbitrary functions which may be helpful to explain some complex phenomena. We also give some figures to describe the property of these solutions. In additions, the method can also be successfully applied to other nonlinear evolution and wave equations.     

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     

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.