广义随机KdV方程新的精确类孤子解

利用厄米(Hermite)变换求出了广义随机KdV方程新的类孤子解.这种方法的基本思想是通过厄米变换把Wick类型的广义随机KdV变成广义变系数KdV方程，利用特殊的截断展开方法求出 方程的解，然后通过厄米的逆变换求出方程的随机解. 关键词： 随机KdV方程 随机孤子解 白色噪音 截断展开方法 厄米变换     

改进的tanh函数方法与广义变系数KdV和MKdV方程新的精确解

利用改进的tanh函数方法将广义变系数KdV方程和MKdV方程化为一阶变系数非线性常微分方 程组-通过求解这个变系数非线性常微分方程组，获得了广义变系数KdV方程和MKdV方程新的 精确类孤子解、有理形式函数解和三角函数解- 关键词： 改进的tanh函数方法 类孤子解 有理形式函数解 三角函数解     

(2+1)-维Wick类型随机广义KP方程的类孤子解

利用埃尔米特变换和特殊的截断展开法求出(2+1)-维Wick类型随机广义KP方程的类孤子解. 这种方法的基本思想是通过埃尔米特变换把(2+1)-维Wick类型随机广义KP方程变成的(2+1)-维广义变系数KP方程,利用特殊的截断展开方法求出方程的解,然后通过埃尔米特的逆变换求出方程的随机解.     

变系数广义KdV方程新的类孤波解和精确解

用普通KdV方程作变换，构造变系数广义KdV方程的解，获得了变系数广义KdV方程新的Jacobi椭圆函数精确解、类孤波解、三角函数解和Weierstrass椭圆函数解. 关键词： KdV方程 变系数广义KdV方程 类孤波解 精确解     

辅助方程构造带强迫项变系数组合KdV方程的精确解

在辅助方程法的基础上给出第一种椭圆辅助方程和函数变换相结合的一种方法,并借助符号计算系统Mathematica构造了带强迫项变系数组合KdV方程的类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 变系数组合KdV方程 精确解     

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     

推广的Painlevé展开及KdV方程的非标准截断解

利用奇性流形的任意性，选用不同的展开函数及非标准截断展开于KdV方程，得到了许多用复杂隐函数表示的精确解. 关键词：     

用试探方程法求变系数非线性发展方程的精确解

将试探方程法应用到变系数非线性发展方程的精确解的求解.以两类变系数KdV方程为例，在相当一般的参数条件下求得了丰富的精确解，其中包括新解. 关键词： 试探方程法 变系数KdV方程 类椭圆正弦(余弦)波解 类孤子解     

变系数非线性发展方程的G＇/G展开解

用近年来提出的（G＇/G）展开法首次尝试了对变系数非线性发展方程的求解,并以两类变系数非线性KdV方程为例,且成功得到了新的精确解.实践证明：（G＇/G）展开法不仅适用于常系数非线性发展方程,而且还很好地适用于变系数非线性方程,具有广泛的应用前景.     

组合KdV方程的显式精确解

借助计算机代数系统Mathematica，利用双曲函数法找到了组合KdV方程(Combined KdV Equation)的精确孤立波解，包括钟型孤立波解和扭结型孤立波解.在此基础上又对双曲函数法的思想进行了推广，从而获得了其更多的显式精确解，包括间断型激波解和指数函数型解.这种方法也适用于求解其他非线性发展方程(组). 关键词： 组合KdV方程 双曲函数法 孤立波解 精确解     

New Exact Solutions to （2＋1）-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Rieatti equation expansion method, we present explicit solutions of the （2＋1）-dimensional variable coefficients Broer-Kaup （VCBK） equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.     

New Exact Solutions to (2+1)-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Ricatti equation expansion method, we present explicit solutions of the (2 1)-dimensional variable coefficients Broer-Kaup (VCBK) equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions.Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.     

Consistent Riccati expansion solvability,symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg-de Vries equation in fluid dynamics of internal solitary waves

We study a forced variable-coefficient extended Korteweg-de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.     

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.     

Explicit Solutions of (2+1)-Dimensional Canonical Generalized KP,KdV, and(2+1)-Dimensional Burgers Equations with Variable Coefficients

In this paper, using the generalized G'/G-expansion method and the auxiliary differential equation method, we discuss the (2+1)-dimensional canonical generalized KP (CGKP), KdV, and (2+1)-dimensional Burgers equations with variable coefficients. Many exact solutions of the equations are obtained in terms of elliptic functions, hyperbolic functions, trigonometric functions, and rational functions.     

带强迫项变系数组合KdV方程的无穷序列复合型类孤子新解

为了获得变系数非线性发展方程的无穷序列复合型新解,研究了G′(ξ)G(ξ)展开法.通过引入一种函数变换,把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题.在此基础上,利用Riccati方程解的非线性叠加公式,获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解.借助这些复合型新解与符号计算系统Mathematica,构造了带强迫项变系数组合KdV方程的无穷序列复合型类孤子新精确解.     

The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term

By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg--de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.     

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.