A Generalized Extended F-Expansion Method and Its Application in (2+1)-Dimensional Dispersive Long Wave Equation

A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the (2 1)-dimensional dispersive long wave equation. With the aid of computerized symbolic computation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained. In the limit cases, the solitary wave solutions are derived as well.     

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the （2＋1）-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.     

A Generalization of F-Expansion Method and Its Application to （2＋l）-Dimensional Boussinesq Equation

A new generalized F-expansion method is introduced and applied to the study of the （2＋1）-dimensional Boussinesq equation. The further extension of the method is discussed at the end of this paper.     

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the (2 1)-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.     

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

The (2+1)-dimensional breaking soliton equation describes the interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. In this paper, with the aid of symbolic computation, six kinds of new special exact soltion-like solutions of (2+1)-dimensional breaking soliton equation are obtained by using some general transformations and the further generalized projective Riccati equation method.     

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 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.     

Extended F-Expansion Method and Periodic Wave Solutionsfor Klein-Gordon-Schrödinger Equations

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrödinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.     

扩展的形变映射方法和(2+1)维破裂孤子方程的新解

扩展的形变映射方法应用于非线性物理模型的研究， 获得了(2+1)维破裂孤子方程的新解, 合适地选择任意函数，可以获得该模型的丰富的局域和非局域的相干结构，这里仅揭示(2+1)维破裂孤子方程(1)和(2)的周期型新孤波结构. 关键词： 形变映射方法 (2+1)维破裂孤子方程 孤波结构     

Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation

Based on the computerized symbolic system Maple and a Riccati equation, a Riccati equation expansion method is presented by a general ansatz. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the Bogoyavlenskii's generalized breaking soliton equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.     

New Generalized Transformation Method and Its Application in Higher-Dimensional Soliton Equation

A new generalized transformation method is presented to find more exact solutions of nonlinear partial differential equation. As an application of the method, we choose the (3+1)-dimensional breaking soliton equation to illustrate the method. As a result many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic function solutions, and rational solutions, are obtained. The new method can be extended to other nonlinear partial differential equations in mathematical physics.     

An Extended Subequation Rational Expansion Method and Solutions of (2+1)-Dimensional Cubic Nonlinear Schrödinger Equation

An extended subequation rational expansion method is presented and used to construct some exact analytical solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation. From our results, many known solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of new non-travelling wave and coefficient function's soliton-like solutions, and elliptic solutions are demonstrated by some plots.     

A Generalized Hirota Ansatz to Obtain Soliton-Like Solutions for a (3+1)-Dimensional Nonlinear Evolution Equation

Instead of the usual Hirota ansatz, i.e., the functions in bilinear equations being chosen as exponential types, a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation. Based on the resulting generalized Hirota ansatz, a family of new explicit solutions for the equation are derived.     

New doubly periodic and multiple soliton solutions of the generalized （3＋l）-dimensional KP equation with variable coefficients

A new generalized Jacobi elliptic function method is used to construct the exact travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new doubly periodic and multiple soliton solutions are obtained for the generalized (3+1)-dimensional Kronig－Penny (KP) equation with variable coefficients. This method can be applied to other equations with variable coefficients.     

An Improved F-Expansion Method and Its Application to Coupled Drinfel'd-Sokolov-Wilson Equation

With the aid of computerized symbolic computation, an improved F-expansion method is presented to uniformly construct more new exact doubly periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear partial differential equations (NPDEs). The coupled Drinfel'd-Sokolov-Wilson equation is chosen to illustrate the method. As a result, we can successfully obtain abundant new doubly periodic solutions without calculating various Jacobi elliptic functions. In the limit cases, the rational solitary wave solutions and trigonometric function solutions are obtained as well.     

On a New Modified Extended Tanh-Function Method

In this paper, a new modified extended tanh-function method is presented for constructing multiple soliton-like, periodic form and rational solutions of nonlinear evolution equations (NLEEs). This method is more powerful thanthe extended tanh-function method [Phys. Lett. A277 (2000) 212] and the modified extended tanh-function method[Phys. Lett. A299 (2002) 179] Abundant new solutions of two physically important NLEEs are obtained by using thismethod and symbolic computation system Maple.     

Abundant Multisoliton Structures of the Generalized Nizhnik-Novikov-Veselov Equation

Using the extended homogenous balance method, we obtain abundant exact solution structures of a (2+1)-dimensional integrable model, the generalized Nizhnik-Novikov-Veselov equation. By means of the leading order term analysis, the nonlinear transformations of generalized Nizhnik-Novikov-Veselov equation are given first, and then some special types of single solitary wave solution and the multisoliton solutions are constructed.     

New Multiple Soliton-like and Periodic Solutions for （2＋l）-Dimensional Canonical Generalized KP Equation with Variable Coefficients

In this paper, the generalized ranch function method is extended to （2＋1）-dimensianal canonical generalized KP （CGKP） equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the （2＋1）-dimensional OGKP equation with variable coetffcients.     

The Peridic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobielliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions andthe other type of traveling wave solutions for the system are obtained.     

Variable Separation Solutions of Generalized Broer-Kaup System via a Projective Method

Using an extended projective method, a new type of variable separation solution with two arbitrary functions of the (2+1)-dimensional generalized Broer-Kaup system (GBK) is derived. Based on the derived variable separation solution, some special localized coherent soliton excitations with or without elastic behaviors such as dromions, peakons, and foldons etc. are revealed by selecting appropriate functions in this paper.