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.     

Auto-Backlund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the extended homogeneous balance method, a new auto-Ba^ecklund transformation(BT) to the generalized Kadomtsew-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.     

A Generalized F-expansion Method and Its Application in High-Dimensional Nonlinear Evolution Equation

A generalized F-expansion method is introduced and applied to （3＋1 ）-dimensional Kadomstev-Petviashvili（KP） equation. As a result, some new Jacobi elliptic function solutions of the equation are found, from which the trigonometric function solutions and the solitary wave solutions can be obtained. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.     

A Generalized Variable-Coefficient Algebraic Method Exactly Solving （3＋1）-Dimensional Kadomtsev-Petviashvilli Equation

A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.     

Solutions to Generalized mKdV Equation

A transformation is introduced for generalized mKdV (GmKdV for short) equation and Jacobi elliptic function expansion method is applied to solve it. It is shown that GmKdV equation with a real number parameter can be solved directly by using Jacobi elliptic function expansion method when this transformation is introduced, and periodic solution and solitary wave solution are obtained. Then the generalized solution to GmKdV equation deduces to some special solutions to some well~known nonlinear equations, such as KdV equation, mKdV equation, when the real parameter is set specific values.     

New Doubly Periodic Solutions of Nonlinear Evolution Equations via Weierstrass Elliptic Function Expansion Algorithm

A Weierstrass elliptic function expansion method and its algorithm are developed in this paper. The method changes the problem solving nonlinear evolution equations into another one solving the corresponding system of nonlinear algebraic equations. With the aid of symbolic computation (e.g. Maple), the method is applied to the combined KdV-mKdV equation and (2 1)-dimensional coupled Davey-Stewartson equation. As a consequence, many new types of doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Jacobi elliptic function solutions and solitary wave solutions are also given as simple limits of doubly periodic solutions.     

The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term

By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg--de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.     

New exact solitary wave solutions to generalized mKdV equation and generalized Zakharov--Kuzentsov equation

In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov--Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.     

Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.     

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.     

On a Generalized Extended F-Expansion Method

Making use of a new generalized ansatz, we present a new generalized extended F-expansion method for constructing the exact solutions of nonlinear partial differential equations in a unified way. Applying the generalized method with the aid of Maple, we consider the (2+1)-dimentional breaking soliton equation. As a result, we successfully obtain some new and more general solutions including Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, and so on. As an illustrative sample, the properties of some soliton solutions for the breaking soliton equation are shown by some figures. Our method can also be applied to other partial differential equations.     

广义Boussinesq方程的无穷序列新精确解

以辅助方程法为基础,给出第二种椭圆方程解的非线性叠加公式,借助符号计算系统Mathematica获得了广义Boussinesq方程的无穷序列新精确解.这里包括无穷序列Jacobi椭圆函数精确解、无穷序列孤立波解和无穷序列三角函数解.该方法在构造非线性发展方程无穷序列精确解方面具有普遍意义.     

A Generalized F-expansion Method and Its Application in High-Dimensional Nonlinear Evolution Equation

A generalized F-expansion method is introduced and applied to (3 1)-dimensional Kadomstev-Petviashvili(KP) equation. As a result, some new Jacobi elliptic function solutions of the equation are found, from which the trigonometric function solutions and the solitary wave solutions can be obtained. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.     

第二种椭圆方程构造变系数非线性发展方程的无穷序列新精确解

本文为了获得非线性发展方程的无穷序列新精确解,进一步研究获得了第二种椭圆方程的几类新型解和Bäcklund变换.在此基础上,借助符号计算系统Mathematica,用带强迫项变系数组合KdV方程、(2+1)维和(3+1)维变系数Zakharov-Kuznetsov 方程为应用实例,构造了无穷序列新精确解.这里包括无穷序列Jacobi 椭圆函数光滑孤立子解、无穷序列Jacobi椭圆函数紧孤立子解、无穷序列三角函数紧孤立子解和无穷序列尖峰孤立子解. 关键词： 第二种椭圆方程 Bä cklund 变换 变系数非线性发展方程 无穷序列新精确解     

Lie Algebras and Integrable Systems

A 3? 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrödinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3? 3 Lie subalgebra into a 2? 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation.     

Solving Nonlinear Wave Equations by Elliptic Equation

The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.     

Solutions to Generalized mKdV Equation

A transformation is introduced for generalized mKdV (GmKdV for short) equation and Jacobi elliptic function expansion method is applied to solve it. It is shown that GmKdV equation with a real number parameter can be solved directly by using Jacobi elliptic function expansion method when this transformation is introduced, and periodic solution and solitary wave solution are obtained. Then the generalized solution to GmKdV equation deduces to some special solutions to some well-known nonlinear equations, such as KdV equation, mKdV equation, when the real parameter is set specific values.     

New Jacobian Elliptic Function Solutions of Modified KdV Equation: Ⅱ

Recently, we obtained thirteen families of Jacobian elliptic function solutions of mKdV equation by usingour extended Jacobian elliptic function expansion method. In this note, the mKdV equation is investigated and anotherthree families of new doubly periodic solutions (Jacobian elliptic function solutions) are fbund again by using a newtransformation, which and our extended Jacobian elliptic function expansion method form a new method still called theextended Jacobian elliptic function expansion method. The new method can be more powertul to be applied to othernonlinear differential equations.     

扩展的Jacobi椭圆函数展开法和Zakharov方程组的新的精确周期解

对Jacobi椭圆函数展开法进行了扩展，且利用这一方法求出了Zakharov方程组的一系列新的精确周期解，在极限情况下可得到相应的孤波解，补充了前面研究的结果. 关键词： Jacobi椭圆函数展开法 非线性发展方程 精确解 周期解