Separation Transformation and New Exact Solutions of the (N+1)-dimensional Dispersive Double sine-Gordon Equation

In this paper,the separation transformation approach is extended to the(N + 1)-dimensional dispersive double sine-Gordon equation arising in many physical systems such as the spin dynamics in the B phase of 3 He superfluid.This equation is first reduced to a set of partial differential equations and a nonlinear ordinary differential equation.Then the general solutions of the set of partial differential equations are obtained and the nonlinear ordinary differential equation is solved by F-expansion method.Finally,many new exact solutions of the(N + 1)-dimensional dispersive double sine-Gordon equation are constructed explicitly via the separation transformation.For the case of N 2,there is an arbitrary function in the exact solutions,which may reveal more novel nonlinear structures in the high-dimensional dispersive double sine-Gordon equation.     

Exact Jacobian Elliptic Function Solutions to sine-Gordon Equation

In this paper, four transformations are introduced to solve single sine-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the single sine-Gordon equation.     

Exact Jacobian Elliptic Function Solutions to sine-Gordon Equation

In this paper, four transformations are introduced to solve single sine-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the single sine-Gordon equation.     

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

New exact solutions expressed by the Jacobi elliptic functions are obtained to the (2+1)-dimensional dispersive long-wave equations by using the modified F-expansion method. In the limit case, new solitary wave solutions and triangular periodic wave solutions are obtained as well.     

Exact Solutions to Degasperis-Procesi Equation

In this paper, dependent and independent variable transformations are introduced to solve the Degasperis-Procesi equation. It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.     

Exact Solutions to Degasperis-Procesi Equation

In this paper, dependent and independent variable transformations are introduced to solve the Degasperis- Procesi equation. It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.     

New Exact Travelling Wave Solutions to Hirota Equation and (1+1)-Dimensional Dispersive Long Wave Equation

Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e.,the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.     

Exact Solutions to Short Pulse Equation

In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.     

Exact Jacobian Elliptic Function Solutions to sinh-Gordon Equation

In this paper, two transformations are introduced to solve sinh-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the sinh-Gordon equation.     

(n+1)维双Sine-Gordon方程的新精确解

给出包含第一种椭圆方程的三角函数型辅助方程及其解的叠加公式.在一般函数变换下,借助符号计算系统Mathematica,构造了(n+1)维双sine-Gordon方程新的Jacobi椭圆函数精确解.这些解包括了行波变换下的Jacobi椭圆函数精确解、精确孤立波解和三角函数解.     

Some New Exact Solutions of Jacobian Elliptic Function of Petviashvili Equation

By using the modified mapping method, we find new exact solutions of the Petviashvili equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.     

Various Types of Kink and Bright Resonant Soliton Solutions for the (2+1)-Dimensional Double sine-Gordon Equation

The algebraic mapping relations between the (2+1)-dimensional double sine-Gordon equation and the cubic nonlinear Klein—Gordon equation are constructed. Many new types of two-dimensional resonant kink, bright soliton and solitoff solutions are obtained, such as broken line shape, "V" shape, "snake" shape and "M" shape solitary waves, Zigzag-curve type, "ω" shape, peroidic-curve type, oscillatory Arch-type and parabolic shape bright soliton waves. We also investigate the propagating properties of some soliton solutions.     

A Bilinear B(a)cklund Transformation and Lax Pair for a (1+1)-Dimensional Differential-Difference sine-Gordon Equation

In this paper, we obtain a 1 1 dimensional integrable differential-difference model for the sine-Gordon equation by Hirota's discretization method. A bilinear B(a)cklund transformation and the associated Lax pair are also proposed for this model.     

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.     

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

Periodic wave solutions to the dispersive long-wave equations are obtained by using the F-expansion method, which can be thought of as a generalization of the Jacobi elliptic function method. In the limit case, solitary wave solutions are obtained as well.     

New Complexiton Solutions of (1+1)-Dimensional Dispersive Long Wave Equation

By means of two different Riccati equations with different parameters as subequation in the components of finite rational expansion method, new complexiton solutions for the (1+1)-dimensional dispersive long wave equation are successfully constructed, which include various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, and various combination of hyperbolic and rational function solutions.     

（2＋1）维sine—Gordon方程的三种函数混合解

基于一构造的Wronskian形式展开法，获得了（2＋1）维sine—Gordon方程的一系列新形式的混合解．这些解是三角函数、双曲函数及Jacobi椭圆函数等三种函数的组合形式．     

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.     

Double Elliptic Equation Expansion Approach and Novel Solutions of （2＋1）-Dimensional Break Soliton Equation

In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the （2＋1）-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in （2＋1） dimensions.     

New Exact Solutions and Dynamics in （3＋1）-Dimensional Gross-Pitaevskii Equation with Repulsive Harmonic Potential

Using an improved homogeneous balance principle and an F-expansion technique, we construct the new exact periodic traveling wave solutions to the （3＋1）-dimensional Gross-Pitaevskii equation with repulsive harmonic potential. In the limit cases, the solitary wave solutions are obtained as well. We also investigate the dynamical evolution of the solitons with a time-dependent complicated potential.