Some new exact solutions to the Burgers-Fisher equation and generalized Burgers-Fisher equation

Some new exact solutions of the Burgers-Fisher equation and generalized Burgers-Fisher equation have been obtained by using the first integral method. These solutions include exponential function solutions, singular solitary wave solutions and some more complex solutions whose figures are given in the article. The result shows that the first integral method is one of the most effective approaches to obtain the solutions of the nonlinear partial differential equations.

Exact Solutions of Two Nonlinear Partial Differential Equations by the First Integral Method

In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software;the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.

New Exact Explicit Solutions of the Generalized Zakharov Equation via the First Integral Method

The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.

Exact Solutions of Generalized Burgers-Fisher Equation with Variable Coefficients

The generalized Riccati equation vational expansion method is extended in this paper. Several exact solutions for the generalized Burgers-Fisher equation with variable coefficients are obtained by this method, and some of which are derived for the first time. It is concluded from the results that this approach is simple and efficient even in solving partial differential equations with variable coefficients.

Trial function method and exact solutions to the generalized nonlinear Schrdinger equation with time-dependent coefficient

In this paper, the trial function method is extended to study the generalized nonlinear Schrodinger equation with time- dependent coefficients. On the basis of a generalized traveling wave transformation and a trial function, we investigate the exact envelope traveling wave solutions of the generalized nonlinear Schrodinger equation with time-dependent coefficients. Taking advantage of solutions to trial function, we successfully obtain exact solutions for the generalized nonlinear Schrodinger equation with time-dependent coefficients under constraint conditions.

A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.

Direct Reduction and Exact Solutions for Generalized Variable Coefficients 2D KdV Equation under Some Integrability Conditions

在同类的平衡(HB ) 之中基于关上的连接方法和 ClarksonKruskal (CK ) 方法，我们学习概括可变系数 2D KdV 方程的类似减小。同时，这在在可变系数之间的一些 integrability 条件下面导致在平常的微分方程形式的直接减小，这被显示出。二个不同盒子被讨论了，为那些平常的微分方程的解决方案的搜索产出在在可变系数之间的一些限制下面的许多准确旅行和在夸张、三角法的函数形式的 solitonic 波浪解决方案。

Extended Complex tanh-Function Method and Exact Solutions to （2＋1）-Dimensional Hirota Equation

在这篇论文，一个新扩大复杂 tanh 功能方法为构造旅行波浪，非旅行的波浪，和系数函数被介绍“非线性的方程的象 soliton 一样解决方案。这个方法比复杂 tanh 功能方法更强大[混乱， Solitons 和分数维图形 20 (2004 ) 1037 ] 。丰富的新答案(2+1 ) 维的 Hirota 方程被使用这个方法和符号的计算系统枫树获得。

New Exact Solutions and Special Coherent Structures for Coupled Integrable Dispersionless Equation

用改进同类的平衡方法，我们少些为联合 integrable 分散获得新准确答案方程。根据这些准确答案，我们由适当地选择任意的功能发现一些新有趣的协调结构。

New Exact Solutions for （2＋1）-Dimensional Breaking Soliton Equation

New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solutionsand triangular periodic wave solutions are obtained.

Symmetry, Reductions and New Exact Solutions of ANNV Equation Through Lax Pair

In this paper,the direct symmetry method is extended to the Lax pair of the ANNV equation.As a result,symmetries of the Lax pair and the ANNV equation are obtained at the same time.Applying the obtained symmetry,the (2+1)-dimensionai Lax pair is reduced to (1+1)-dimensional Lax pair,whose compatibility yields the reduction ofthe ANNV equation.Based on the obtained reductions of the ANNV equation,a lot of new exact solutions for theANNV equation are found.This shows that for an integrable system,both the symmetry and the reductions can beobtained through its corresponding Lax pair.

Nonequivalent Similarity Reductions and Exact Solutions for Coupled Burgers-Type Equations

Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations.The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions,hyperbolic functions,and trigonometric functions.Some figures are given to show the properties of the solutions.

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified.

Two Classes of New Exact Solutions to （2＋1）-Dimensional Breaking Soliton Equation

The singular manifold method is used to obtain two general solutions to a (2+1)-dimensional breaking soliton equation, each of which contains two arbitrary functions. Then the new periodic wave solutions in terms of the Jacobi elliptic functions are generated from the general solutions. The long wave limit yields the new types of dromion and solitary structures.

The First Integral Method to Study a Class of Reaction-Diffusion Equations

In this letter, a class of reaction-diffusion equations, which arise in chemical reaction or ecology and other fields of physics, are investigated. A more general analytical solution of the equation is obtained by using the first integral method.

New Exact Solutions to Dispersive Long-Wave Equations in （2＋1）-Dimensional Space

Jacobi 椭圆形的函数表示的新准确解决方案被获得到(2+1 ) 由使用修改 F 扩大方法的维的散 Jong 波浪方程。Inthe 限制大小写，新独居的波浪答案和三角形的周期的波浪答案也被获得。

On an Auto-Bācklund Transformation for （2＋1）-Dimensional Variable Coefficient Generalized KP Equations and Exact Solutions

By the application of the extended homogeneous balance method, we derive an auto-Bācklund transformation (BT) for (2＋1)-dimensional variable coefficient generalized KP equations. Based on the BT, in which there are two homogeneity equations to be solved, we obtain some exact solutions containing single solitary waves.

A class of exact solutions for N-dimensional incompressible magnetohydrodynamic equations

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic （MHD） equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations axe directly constructed.