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.     

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.     

Solitary wave solutions of two nonlinear physical models by tanh–coth method

In this work, we established the exact solutions for some nonlinear physical models. The tanh–coth method was used to construct solitary wave solutions of nonlinear evolution equations. The tanh–coth method presents a wider applicability for handling nonlinear wave equations.     

New travelling wave solutions to the Boussinesq and the Klein–Gordon equations

In this work, many new travelling wave solutions are established for the Boussinesq and the Klein–Gordon equations. The extended tanh method, the rational hyperbolic functions method, and the rational exponential functions method are used to generate these new solutions. The new solutions are bell-shaped solitons, periodic, and complex solutions. The proposed approaches are also applicable to a large variety of nonlinear evolution equations.     

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 exact solutions to some variable coefficients problems

With the aid of computer symbolic computation system such as Maple, an extended tanh method is applied to determine the exact solutions for some nonlinear problems with variable coefficients. Several new soliton solutions and periodic solutions can be obtained if we taking paraments properly in these solutions. The employed methods are straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical 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.     

Extending the generalized tanh method to the generalized Hamiltonian equations: New soliton-like solutions

To certain nonlinear evolution equations, the tanh method has been generalized for constructing not only solitary-wave but also soliton-like solutions. In this paper, no loss of conciseness, we further extend the generalized tanh method with computerized symbolic computation to a pair of generalized Hamiltonian equations. A new family of soliton-like analytical solutions is obtained, of which the solitary waves and previously-claimed soliton-like solutions are shown to be the special cases.     

The extended tanh method for solving systems of nonlinear wave equations

The extended tanh method with a computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks and plane periodic solutions. The applied method will be used to solve the generalized coupled Hirota Satsuma KdV equation.     

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.     

Some Exact Solutions of Two Fifth Order KdV-type Nonlinear Partial Differential Equations

We consider the generalized integrable fifth order nonlinear Korteweg-de Vries (fKdV) equation. The extended Tanh method has been used rigorously, by computational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.     

求解非线性方程的双函数法

基于齐次平衡法和李志斌的tanh函数法,得到简单有效的求解非线性发展方程的双函数法,这种方法利用非线性发展方程孤立波的局部性特点,把非线性方程的孤波解表示为函数f和g的多项式,并用这种方法求出了非线性波理论中的基本模型KdV方程的多组孤波解。     

一类求行波解的线性方法

基于齐次平衡法和李志斌的 tanh函数法 ,本文得到一类简单有效的求解非线性发展方程的线性方法 .这类方法利用非线性发展方程孤立波的局部性特点 ,适当地选取函数 f 和 g,将孤波表示为 f,g的多项式 ,从而将非线性发展方程求解问题转化为非线性代数方程组的求解问题 ,再利用吴消元法求解方程组从而得到非线性发展方程的行波解     

Solitary solution of modified bad and good Boussinesq equation by using of tanh and extended tanh methods

In this work, a mathematical approach based on the reduction of order for solving ordinary differential equations, the standard tanh method and the extended tanh method has been used to obtain solutions of modified bad Boussinesq and modified good Boussinesq equations.     

Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics

In this paper, we devise a new unified algebraic method to construct a series of explicit exact solutions for general nonlinear equations. Compared with most existing methods such as tanh method, Jacobi elliptic function method and homogeneous balance method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic, and soliton solutions, (f) Jacobi, and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, combined KdV–MKdV, Camassa–Holm, Kaup–Kupershmidt, Jaulent–Miodek, (2+1)-dimensional dispersive long wave, new (2+1)-dimensional generalized Hirota, (2+1)-dimensional breaking soliton and double sine-Gordon equations. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.     

Soliton-like and periodic form solutions to (2 + 1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE, we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.     

An algebraic method exactly solving two high-dimensional nonlinear evolution equations

An algebraic method is applied to construct soliton solutions, doubly periodic solutions and a range of other solutions of physical interest for two high-dimensional nonlinear evolution equations. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solutions according to some parameters.     

A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation

In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.     

Symbolic computation and a uniform direct ansätze method for constructing travelling wave solutions of nonlinear evolution equations

On the basis of the computer symbolic system Maple and the tanh method, the Riccati equation method as well as all kinds of improved versions of these methods, we present a further uniform direct ansätze method for constructing travelling wave solutions of nonlinear evolution equations. Compared with the existing methods, the presented method can be used to construct more new general solutions. And we give some examples to illustrate the key step of our method.     

The extended tanh method for the Zakharov–Kuznetsov (ZK) equation,the modified ZK equation,and its generalized forms

The reliable extended tanh method, that combines tanh with coth, is used for analytic treatment of the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation, and the generalized forms of these equations. New travelling wave solutions with solitons and periodic structures are determined. The power of the employed method is confirmed.