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.     

非线性波动方程的孤波解

用平衡法并结合吴消元法得到了一类较广泛非线性波动方程u_tt-a₁u_xx+a₂u_t+a₃u+a₄u³=0的若干孤波解公式，从而物理学上许多著名的方程，如φ⁴方程、Klein-Gordon方程、Landau-Ginzburg-Higgs方程、非线性电报方程等都可作为该方程的特殊情形得到相应的孤波解 关键词：     

Solution of ODE u＂ ＋ p（u）（u＇）^2＋ q（u） = 0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u＂ ＋p（u）（u＇）^2 ＋ q（u） = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u＂＋p（u）（u＇）^2＋q（u） = 0 includes the equation （u＇）^2 = f（u） as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.     

Extended Jacobi elliptic function method and its applications to （2＋l）-dimensional dispersive long-wave equation

An extended Jacobi elliptic function method is proposed for constructing the exact double periodic solutions of nonlinear partial differential equations (PDEs) in a unified way. It is shown that these solutions exactly degenerate to the many types of soliton solutions in a limited condition. The Wu－Zhang equation (which describes the (2+1)-dimensional dispersive long wave) is investigated by this means and more formal double periodic solutions are obtained.     

New Approach to Find Exact Solutions to Classical Boussinesq System

In this paper, based on a new system of three Riccati equations, we give a new method to construct more new exact solutions of nonlinear differential equations in mathematical physics. The classical Boussinesq system is chosen to illustrate our method. As a consequence, more families of new exact solutions are obtained, which include solitary wave solutions and periodic solutions.     

Applications of Elliptic Equation to Nonlinear Coupled Systems

The elliptic equation is taken as a transformation and applied to solve nonlinear coupled systems. 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 this method can be taken as a unified method in solving nonlinear coupled systems.     

On High-Frequency Soliton Solutions to a （2＋1）-Dimensional Nonlinear Partial Differential Evolution Equation

A （2＋1）-dimensional nonlinear partial differential evolution （NLPDE） equation is presented as a model equation for relaxing high-rate processes in active barothropic media. With the aid of symbolic computation and Hirota＇s method, some typical solitary wave solutions to this （2＋1）-dimensional NLPDE equation are unearthed. As a result, depending on the dissipative parameter, single and multivalued solutions are depicted.     

New Types of Exact Solutions for （N＋1）-Dimensional φ^4 Model

New types of exact solutions of the （N ＋ 1）-dimensional φ^4-model are studied in detail. Some types of interaction solutions such as the periodic-periodic interaction waves and the periodic-solitary wave interaction solutions are found.     

The Periodic Wave Solutions for Two Nonlinear Evolution Equations

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

Solitary Wave Solutions of Triple Sine-Gordon Equation

Through a mapping relation between the triple sine-Gordon and Ф5 model, we discuss the equation in detail. A series of solitary wave solutions for the triple sine-Gordon equation is obtained by the existing ones of the Ф5 model. Some detailed solitary wave solutions are given for some special coefficients which have significance in physics.     

一类非线性方程的新周期解

把Jacobi椭圆函数展开法扩展到Jacobi椭圆余弦函数和第三类Jacobi椭圆函数的有限展开法,并给出了一类非线性波动方程的新周期解,并且应用这种方法得到的周期解也可以退化为冲击波解或孤波解. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤波解     

Combined periodic wave and solitary wave solutions in two-component Boseben Einstein condensates

In this paper, the Jacobi elliptic function expansion method provides an effective approach to obtain the exact periodic wave solutions of two-component Bose-Einstein condensates. Exact combined bright-bright and dark-dark soliton wave solutions can be achieved in their limit conditions. We also obtain the different formation regions of combined solitons. Our results show that the intraspecies (interspecies) interaction strengths clearly affect the formation of dark-dark, bright-bright and dark-bright soliton solutions in different regions.     

非线性波动方程的Jacobi椭圆函数包络周期解

应用Jacobi椭圆函数展开法求得了一类非线性波方程的包络周期解,而且用这种方法得到的周期解在一定条件下可以退化为包络冲击波解或包络孤立波解 关键词： Jacobi椭圆函数 非线性方程 包络周期解 包络孤立波解     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and 0, the hyperbolic function solutions (including the solitary wave solutions) and trigonometric solutions are also given respectively.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and O, the hyperbolic function solutions （including the solitary wave solutions） and trigonometric solutions are also given respectively.     

立方非线性Schr?dinger方程的Weierstrass椭圆函数周期解

利用Weierstrass椭圆函数展开法对非线性光学、等离子体物理等许多系统中出现的立方非线性 Schr(o)dinger方程进行了研究.首先通过行波变换将方程化为一个常微分方程,再利用Weierstrass椭圆函数展开法思想将其化为一组超定代数方程组,通过解超定方程组,求得了含Weierstrass椭圆函数的周期解,以及对应的Jacobi椭圆函数解和极限情况下退化的孤波解.该方法有以下两个特点:一是可以借助数学软件Mathematica自动地完成;二是可以用于求解其它的非线性演化方程(方程组).     

Applications of F-expansion method to the coupled KdV system

An extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics is presented, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by Jacobi elliptic functions for the coupled KdV equations are derived. In the limit cases, the solitary wave solutions and the other type of travelling wave solutions for the system are also obtained.     

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.     

Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations

In this paper, the improved Jacobian elliptic function expansion approach is extended and applied to constructing discrete solutions of the semi-discrete coupled modified Korteweg de Vries (mKdV) equations with the aid of the symbolic computation system Maple. Some new discrete Jacobian doubly periodic solutions are obtained. When the modulus $m \rightarrow 1$, these doubly periodic solutions degenerate into the corresponding solitary wave solutions, including kink-type, bell-type and other types of excitations.     

非线性波方程求解的新方法

从Legendre椭圆积分和Jacobi椭圆函数的定义出发，得到了新的变换，并把它用于非线性演化方程的求解.用三个具体的例子，如非线性Klein-Gordon方程、Boussinesq方程和耦合的mKdV方程组，说明了具体的求解步骤.比较方便地得到非线性演化方程或方程组的新解析解，如周期解、孤子解等. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤子解