一类高维耦合的非线性演化方程的简单求解

利用一个简单的变换，一类高维耦合的非线性演化方程可以被约化为一低维的简单方程，将已有的求解法应用于简单方程，十分简捷的获得了原方程大量的精确解. 关键词： 非线性耦合方程 精确解 tanh函数方法     

双曲函数型辅助方程构造具5次强非线性项的波方程的新精确孤波解

在辅助方程法的基础上利用两种函数变换和一种双曲函数型辅助方程，通过符号计算系统Mathematica构造了在力学当中一个重要的模型，有5次强非线性项的波方程的新三角函数型和双曲函数型精确孤波解.这种方法寻找其他具5次强非线性项的非线性发展方程的新精确解方面具有普遍意义. 关键词： 双曲函数型辅助方程 函数变换 具5次强非线性项的波方程 精确孤波解     

BBM方程和修正的BBM方程新的精确孤立波解

采用一种双曲函数假设和一类新的辅助常微分方程相结合的方法给出BBM方程和修正的BBM 方程新的精确孤立波解.这种方法也可用于寻找其他非线性发展方程新的孤立波解. 关键词： 辅助方程 双曲函数假设 孤立波解     

非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

构造非线性发展方程精确解的一种方法

在双曲正切函数法、齐次平衡法、辅助方程法的基础上引入非线性发展方程的一个新形式解和新辅助方程，并利用符号计算系统Mathematica构造了Benjamin-Bona-Mahoney(BBM)方程和修正的 BBM方程的新精确孤立波解.这种方法在寻找其他非线性发展方程的新精确解方面具有普遍意义. 关键词： 新辅助方程 形式解 非线性发展方程 精确孤立波解     

扩展齐次平衡法与Backlund变换

将求解非线性演化方程的齐次平衡法进行了扩展,使其包含一个任意函数.此改进方法可得到耦合KdV-Burgers方程、KdV-Burgers方程、Boussinesq方程和一般KdV方程等许多非线性演化方程的Backlund变换和新的精确解.     

非线性Schrdinger方程的包络形式解

扩展了最近提出的F展开方法以构造非线性演化方程更多的精确解,即将F展开法中的一阶非线性常微分方 程和单变量的有限幂级数代之以类似的一阶常微分方程组和两个变量的有限幂级数,这两个变量是一阶常微分方 程组的解分量.作为例子,用扩展的F展开法解非线性Schr dinger方程,得到了很丰富的包络形式的精确解,特别 是以两个不同的Jacobi椭圆函数表示的解.显然,扩展的F展开方法也可以解其他类型的非线性演化方程.     

构造变系数非线性发展方程精确解的一种方法

给出构造变系数非线性发展方程精确解的一种函数变换,并和第二种椭圆方程相结合,借助符号计算系统Mathematica,以带强迫项变系数组合KdV方程为例,得到了该方程新的类Jacobi椭圆函数精确解以及退化后的类孤子解和三角函数解. 关键词： 辅助方程 函数变换 变系数非线性发展方程 精确解     

同伦分析法在求解非线性演化方程中的应用

利用同伦分析法求解了(2+1)维改进的 Zakharov-Kuznetsov方程, 得到了它的近似周期解,该解与精确解符合很好. 结果表明，同伦分析法在求解高维非线性演化方程时, 仍然是一种行之有效的方法. 同时,还对该方法进行了一定的扩展， 经过扩展后的方法能够更方便地求解更多非线性演化方程的高精度近似解析解. 关键词： 同伦分析法 改进的 Zakharov-Kuznetsov方程 周期解     

Lam函数和非线性演化方程的多级准确解的不变性

利用小扰动方法对非线性演化方程作展开得到原始方程的各级近似方程.并在Lam方程和Lam函数的基础上，应用 Jacobi椭圆函数展开法求出了非线性演化方程的多级准确解，从而得到了多级准确解中存在的守恒形式. 关键词： Lam函数 Jacobi椭圆函数 多级准确解 非线性演化方程 扰动方法 不变性     

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.     

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     

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.     

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding Bäcklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.     

一类非线性演化方程的新多级准确解

在Lamé方程和新的Lamé函数的基础上，应用小扰动方法和Jacobi椭圆函数展开法求解一类非线性演化方程（如mKdV方程，非线性Klein-Gordon方程Ⅱ等），获得多种新的多级准确解 .这些多级准确解对应着不同形式的周期波解.这些解在极限条件下可以退化为多种形式的孤 立波解，如带状孤立子、钟形孤立子等. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

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 ³He 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.     

Lam函数和非线性演化方程的扰动方法

利用小扰动方法对非线性演化方程作展开得到原始方程的各级近似方程.应用Jacobi椭圆函 数展开法求得了零级近似方程的准确解，并由此得到一级近似方程和二级近似方程分别满足 齐次Lam方程和非齐次Lam方程，应用Lam函数和Jacobi椭圆函数展开法可以分别求得一级近似方程和二级近似方程的准确解.这样，就求得了非线性演化方程的多级准确解. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

圆杆波导中的一个非线性波动方程及准确周期解

在小变形条件下，采用Cox的非线性应力应变关系，计及横向Possion效应，借助Hamilton变分原理导出了非线性弹性圆杆波导中的纵向波动方程. 利用Jacobi椭圆余弦函数展开法，对该方程与截断的非线性波动方程进行求解，得到了两类非线性波动方程的准确周期解，它们可以进一步退化为孤波解. 关键词： 非线性波 Possion效应 Jacobi椭圆余弦函数     

New Modified Jacobi Elliptic Function Expansion Method and Its Application to (3+1)-Dimensional KP Equation

With the aid of computerized symbolic computation, the new modified Jacobi elliptic function expansion method for constructing exact periodic solutions of nonlinear mathematical physics equation is presented by a new general ansatz. The proposed method is more powerful than most of the existing methods. By use of the method, we not only can successfully recover the previously known formal solutions but also can construct new and more general formal solutions for some nonlinear evolution equations. We choose the (3+1)-dimensional Kadomtsev-Petviashvili equation to illustrate our method. As a result, twenty families of periodic solutions are obtained. Of course, more solitary wave solutions, shock wave solutions or triangular function formal solutions can be obtained at their limit condition.     

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.