Applications of Extended Mapping Deformation Method in Two (3+1)-Dimensional Nonlinear Models

Based on the extended mapping deformation method and symbolic computation, many exact travelling wave solutions are found for the (3+1)-dimensional JM equation and the (3+1)-dimensional KP equation. The obtained solutions include solitary solution, periodic wave solution, rational travelling wave solution, and Jacobian and Weierstrass function solution, etc.     

The extended (G′/G)-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law non-linearity by the extended (G′/G)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.     

New Types of Travelling Wave Solutions From (2+1)-Dimensional Davey-Stewartson Equation

In this paper, based on new auxiliary nonlinear ordinary differential equation with a sixth-degree nonlinear term, we study the (2 1)-dimensional Davey-Stewartson equation and 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.     

New Types of Travelling Wave Solutions From （2＋l）-Dimensional Davey-Stewartson Equation

In this paper, based on new auxiliary nonlinear ordinary differential equation with a sixtb-aegree nonnneal term, we study the （2＋l ）-dimensional Davey-Stewartson equation and 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.     

New travelling wave solutions for combined KdV-mKdV equation and （2＋1）-dimensional Broer- Kaup- Kupershmidt system

Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji {\it et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg--de Vries (KdV) and modified KdV (mKdV) equation and (2+1)-dimensional Broer--Kaup--Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.     

Explicit and exact travelling plane wave solutions of the （2＋1）—dimensional Boussinesq equation

The deformation mapping method is applied to solve a system of (2+1)-dimensional Boussinesq equations. Many types of explicit and exact travelling plane wave solutions, which contain solitary wave solutions,periodic wave solutions,Jacobian elliptic function solutions and others exact solutions, are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and the cubic nonlinear Klein－Gordon equation.     

A General Mapping Approach and New Travelling Wave Solutions to (2+1)-Dimensional Boussinesq Equation

A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.     

Bifurcations and new exact travelling wave solutions for the bidirectional wave equations

By using the method of dynamical system, the bidirectional wave equations are considered. Based on this method, all kinds of phase portraits of the reduced travelling wave system in the parametric space are given. All possible bounded travelling wave solutions such as dark soliton solutions, bright soliton solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, numerical simulations are conducted for dark soliton solutions, bright soliton solutions and periodic travelling wave solutions to the bidirectional wave equations. The results presented in this paper improve the related previous studies.     

Exact Solutions to (2+1)-Dimensional Kaup--Kupershmidt Equation

In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the (2+1)-dimensional Kaup-Kupershmidt (KK) equation are presented. We successfully obtain more general exact travelling wave solutions for (2+1)-dimensional KK equation by the symmetry method and the (G^, /G)-expansion 　method. Consequently, we find some new solutions of (2+1)-dimensional KK equation, 　including similarity solutions, solitary wave solutions, and 　periodic solutions.     

New exact travelling wave solutions of the generalized zakharov equations

A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained.     

Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.     

Travelling Wave Solutions to a Special Type of Nonlinear Evolution Equation

A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of "rank". The key idea of this method is to make use of the arbitrariness of the manifold in Painleve analysis. We selected a new expansion variable and thus obtained a rich variety of travelling wave solutions to nonlinear evolution equation, which covered solitary wave solutions, periodic wave solutions, Weierstrass elliptic function solutions, and rational solutions. Three illustrative equations are investigated by this means, and abundant travelling wave solutions are obtained in a systematic way. In addition, some new solutions are firstly reported here.     

A GROUP OF EXACT (3+1) DIMENSIONAL CYLINDRICALLY SYMMETRIC WAVE SOLUTIONS TO THE EINSTEIN GRAVITATIONAL FIELD EQUATION

Starting with the diagonal spacetime metric tensor, the Einstein gravitational field equation is solved, and a set of exact (3+1) dimensional cylindrically symmetric wave solutions with two arbitrary functions are found. In these solutions all nonvanishing components of spacetime metric tensor are varying with the same propagating factor (ct-z) while the waves are travelling along z axis. The physical picture and the condition of positive energy density of the wave solutions are discussed.     

Travelling wave solutions for (<Emphasis Type="Italic">N</Emphasis> + 1)-dimensional nonlinear evolution equations

In this paper, we implement the exp-function method to obtain the exact travelling wave solutions of (N + 1)-dimensional nonlinear evolution equations. Four models, the (N + 1)-dimensional generalized Boussinesq equation, (N + 1)-dimensional sine-cosine-Gordon equation, (N + 1)-double sinh-Gordon equation and (N + 1)-sinhcosinh-Gordon equation, are used as vehicles to conduct the analysis. New travelling wave solutions are derived.     

非线性Klein-Gordon方程新的精确解

将行波变换替换为更一般的函数变换,推广了修正的Jacobi椭圆函数展开方法.给出了非线性 Klein-Gordon方程新的周期解.当模m→1或m→0时,这些解退化成相应的孤立波解、三 角函数解和奇异的行波解.对于某些非线性方程,在一定条件下一般变换退化为行波约化. 关键词： Jacobi椭圆函数 非线性发展方程 精确解     

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.     

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.     

Variable-Coefficient Hyperbola Function Method and Its Application to （2＋1）-Dimensional Variable-Coefficient Broer-Kaup System

Based on a new intermediate transformation, a variable-coeFficient hyperbola function method is proposed.Being concise and straightforward, it is applied to the (2 1)-dimensional variable-coeFficient Broer-Kaup system. As a result, several new families of exact soliton-like solutions are obtained, besides the travelling wave. When imposing some conditions on them, the new exact solitary wave solutions of the (2 1)-dimensional Broer Kaup system are given. The method can be applied to other variable-coeFficient nonlinear evolution equations in mathematical physics.     

Variable-Coefficient Hyperbola Function Method and Its Application to (2+1)-Dimensional Variable-Coefficient Broer-Kaup System

Based on a new intermediate transformation, a variable-coefficient hyperbola function method is proposed.Being concise and straightforward, it is applied to the (2 1)-dimensional variable-coefficient Broer Kaup system. As a result, several new families of exact soliton-like solutions are obtained, besides the travelling wave. When imposing some conditions on them, the new exact solitary wave solutions of the (2 1)-dimensional Broer-Kaup system are given. The method can be applied to other variable-coefficient nonlinear evolution equations in mathematical physics.     

Linear superposition method for （2＋1）-dimensional nonlinear wave equations

New forms of different-periodic travelling wave solutions for the (2+1)-dimensional Zakharov--Kuznetsov (ZK) equation and the Davey--Stewartson (DS) equation are obtained by the linear superposition approach of Jacobi elliptic function. A sequence of cyclic identities plays an important role in these procedures.