A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

Based on the homogeneous balance method,the Jacobi elliptic expansion method and the auxiliary equation method,the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations.New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple.The method is also valid for other (1+1)-dimensional and higher dimensional systems.     

New Explicit and Exact Solutions for the Klein-Gordon-Zakharov Equations

In this paper, based on the generalized Jacobi elliptic function expansion method, we obtain abundant new explicit and exact solutions of the Klein-Gordon- Zakharov equations, which degenerate to solitary wave solutions and triangle function solutions in the limit cases, showing that this new method is more powerful to seek exact solutions of nonlinear partial differential equations in mathematical physics.     

The Riccati Equation Method Combined with the Generalized Extended $(G'/G)$-Expansion Method for Solving the Nonlinear KPP Equation

An analytic study of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation is presented in this paper. The Riccati equation method combined with the generalized extended $(G''/G)$-expansion method is an interesting approach to find more general exact solutions of the nonlinear evolution equations in mathematical physics. We obtain the traveling wave solutions involving parameters, which are expressed by the hyperbolic and trigonometric function solutions. When the parameters are taken as special values, the solitary and periodic wave solutions are given. Comparison of our new results in this paper with the well-known results are given.     

Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method

The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov （KPP） equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

耦合KdV和Schwarzian KdV方程的显式行波解(英文)

In this paper the ( G’/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters are obtained, which are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are derived from the travelling waves. The (G’/G)-expansion method presents a wider applicability for handling nonlinear wave equations.     

Double Traveling Wave Solutions of the Coupled Nonlinear Klein-Gordon Equations and the Coupled Schrdinger-Boussinesq Equation

The new multiple(G′/G)-expansion method is proposed in this paper to seek the exact double traveling wave solutions of nonlinear partial differential equations.With the aid of symbolic computation,this new method is applied to construct double traveling wave solutions of the coupled nonlinear Klein-Gordon equations and the coupled Schrdinger-Boussinesq equation.As a result,abundant double traveling wave solutions including double hyperbolic tangent function solutions,double tangent function solutions,double rational solutions,and a series of complexiton solutions of these two equations are obtained via this new method.The new multiple(G′/G)-expansion method not only gets new exact solutions of equations directly and effectively,but also expands the scope of the solution.This new method has a very wide range of application for the study of nonlinear partial differential equations.     

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.     

A boundary expansion of solutions to nonlinear singular elliptic equations

In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary.This expansion formula shows the singularity profile of solutions at the boundary.We deal with both linear and nonlinear elliptic equations,including fully nonlinear elliptic equations and equations of Monge-Ampère type.     

Propagation of traveling wave solutions for nonlinear evolution equation through the implementation of the extended modi ed direct algebraic method

In this work,di erent kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional(3-D)nonlinear evolution equations(NEEs)through the implementation of the modi ed extended direct algebraic method.Bright-singular and dark-singular combo solitons,Jacobi's elliptic functions,Weierstrass elliptic functions,constant wave solutions and so on are attained beside their existing conditions.Physical interpretation of the solutions to the 3-D modi ed KdV-Zakharov-Kuznetsov equation are also given.     

Generalized dromions of the (2+1)-dimensional nonlinear Schrdinger equations

IntroductionMost of the mathematical work in the realm of nonlinear phenomena refers to integrablenonlinear equation and their exact soluted. The existence of more generalized locajized solutions for the (2+l)-dimensional KdV ~boil] and the (2+1)-dimensional breaking solitonequationl'] apart from the basic dro~ ~.sl3'41 has given an impetus to search for amore general class of localized sol~ in Oafs (2+1)-dimensional nonlinear evolution equations. Recelltlyt stating from the symmeq constraint…     

Rational Jacobi elliptic solutions for nonlinear differential–difference lattice equations

In this work, we present a direct new method for constructing the rational Jacobi elliptic solutions for nonlinear differential–difference equations, which may be called the rational Jacobi elliptic function method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential–difference equations in mathematical physics via the lattice equation. The proposed method is more effective and powerful for obtaining the exact solutions for nonlinear differential–difference equations.     

A note on the elliptic equation method

The elliptic equation method is improved for constructing exact travelling wave solutions of nonlinear partial differential equations (PDEs). The rational forms of Jacobi elliptic functions are presented. By using new Jacobi elliptic function solutions of the elliptic equation, new doubly periodic solutions are obtained for some important PDEs. This method can be applied to many other nonlinear PDEs.     

Lamé函数和非线性演化方程的新多级准确解

基于Lamé方程和新的Lamé函数,应用摄动方法和Jacobi椭圆函数展开法求解非线性演化方程,获得多种新的多级准确解.这些解在极限条件下可以退化为各种形武的孤波解.     

五阶变系数模型方程新的Jacobi椭圆函数解

我们给出了一种统一的Jacobi椭圆函数方法来构造非线性偏微分方程精确行波解的新方法．借助于Mathematica，我们获得了五阶变系数模型方程的24种Jacobi椭圆函数解．     

Exact solutions of coupled nonlinear Klein-Gordon equation

In this paper, we employ the general integral method for traveling wave solutions of coupled nonlinear Klein-Gordon equations. Based on the idea of the exact Jacobi elliptic function, a simple and efficient method is proposed for obtaining exact solutions of nonlinear evolution equations. The solutions obtained include solitons, periodic solutions and Jacobi elliptic function solutions.     

More solutions of the auxiliary equation to get the solutions for a class of nonlinear partial differential equations

In this paper, a new auxiliary function method is presented for constructing exact travelling wave solutions of nonlinear evolution equations. By the relationships of Jacobi elliptic functions, we get more solutions of the auxiliary equation compared with El-Wakila and Abdou (2006) [22]. So, more new exact travelling wave solutions are obtained for a class of nonlinear partial differential equations.     

用改进的代数方法构造(2+1)维ZK-MEW方程的精确行波解

利用一种改进的统一代数方法将构造(2+1)维ZK MEW((2+1)-dimensionalZakharov-Kuznetsovmodifiedequalwidth)方程精确行波解的问题转化为求解一组非线性的代数方程组．再借助于符号计算系统Mathematica求解所得到的非线性代数方程组,最终获得了方程的多种形式的精确行波解．其中包括有理解,三角函数解,双曲函数解,双周期Jacobi椭圆函数解,双周期Weierstrass椭圆形式解等．并给出了部分解的图形.     

A New Jacobi Elliptic Function Expansion Method for Solvinga Nonlinear PDE Describing Pulse Narrowing Nonlinear Transmission Lines

In this article, we apply the first elliptic function equation to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing pulse narrowing nonlinear transmission lines are given with the aid of computer program, e.g. Maple or Mathematica. Based on Kirchhoff's current law and Kirchhoff's voltage law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and can be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.     

New generalized Jacobi elliptic function expansion method

A new generalized Jacobi elliptic function expansion method is described and used for constructing many new exact travelling wave solutions for nonlinear partial differential equations (PDEs) in a unified way. We obtain many new Jacobi and Weierstrass double periodic elliptic function solutions for (3 + 1)-dimensional Kadmtsev–Petviashvili (KP) equation. This method can be applied to many other equations.