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     

截断展开方法和广义变系数KdV方程新的精确类孤子解

利用特殊的截断展开方法求出了广义变系数KdV方程新的类孤子解.这种方法的基本思想是假定形式解具有截断展开形式,以致可把广义变系数KdV方程转化为一组待定函数的代数方程组,进而给出待定函数容易积分的常微分方程.利用例子证明了这种方法是十分有效的. 关键词： 截断展开法 变系数 KdV方程 孤波解     

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     

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     

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     

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.     

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.     

第二类变系数KdV方程的新类型无穷序列精确解

为了构造变系数非线性发展方程的无穷序列新精确解, 发掘第一种椭圆辅助方程的构造性和机械化性特点, 获得了该方程的 新类型解和相应的 Bäcklund 变换. 在符号计算系统 Mathematica 的帮助下, 以第二类变系数 KdV 方程为应用实例, 构造了三种类型的无穷序列新精确解. 这里包括无穷序列光滑类孤子解、无穷序列尖峰孤立子解和无穷序列紧孤立子解. 这种方法也可以获得其他变系数非线性发展方程的无穷序列新精确解.     

带强迫项变系数组合KdV方程的显式精确解

通过构造两个新的Riccati方程组，推广了Riccati方法，使其具有简洁的形式，丰富和发展了已有的结果，借助Mathematica软件，求出了带强迫项变系数组合KdV方程的一些精确解，包括各种类孤波解、类周期解和变速孤波解. 关键词： Riccati方程组 变系数组合KdV方程 强迫项 类孤波解     

Classification of Dark Modified KdV Equation

The dark Korteweg-de Vries(KdV) systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV(MKdV) systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Furthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The recursion operators of two classes of dark MKdV systems are also given.     

On the uniqueness of Bäcklund transformations for KdV-type equations

Conclusions The restricted Bäcklund transformations for the KdV and MKdV equation are uniquely determined up to two real parameters. One of them can be incorporated into the solutions of the equations and is, therefore, inessential. The only possible generalization of the Bäcklund transformations for the KdV and MKdV equations which can be obtained is through enlarging the number of independent variables of the functions P, Q, i.e. through extending the dependence of the Bäcklund transformations so that they include z_xy, z_yy and possibly higher derivatives. The derivation of these more general Bäcklund transformation by the Clairin's method seems to be very difficult.Theorem 1 was found to be a powerful tool for the proof of the uniqueness of the restricted Bäcklund transformations for the KdV and MKdV equations. Actually, it was used only for very special cases of its range of validity, and it seems that it may be useful for the investigation of the class of the KdV-type equations which possess a restricted Bäcklund transformation. We hope to publish the results of that investigation in the near future.The origin of these investigations on Bäcklund transformations stems from a seminar held with Stanly Steinberg and Kurt Bernardo Wolf. The author is grateful to both of them for his introduction to the problem.The author would also like to thank IIMAS of the National University of Mexico for the hospitality extended to him.     

2N+1阶KdV型方程的孤波解

获得了２Ｎ＋１阶ＫｄＶ型方程的显式精确孤波解．作为特例，讨论了高阶广义ＫｄＶ型方程、高阶广义ＭＫｄＶ型方程和高阶广义Ｓｃｈａｍｅｌ的ＭＫｄＶ型方程．还研究了２Ｎ＋１阶ＫＰ型方程 关键词：     

Bäcklund transformations for the cylindrical Korteweg-de Vries equation

In this letter we derive a Bäcklund transformation (BT) for the cylindrical KdV equation, show that this equation is the only one of a certain class possessing a BT and derive a second BT from the first. Finally we investigate the existence of BTs for generalized MKdV equations.     

The Hamiltonian Canonical Form for Euler-Lagrange Equations

Based on the theory of calculus of variation, some suffcient conditions are given for some Euler-Lagrangcequations to be equivalently represented by finite or even infinite many Hamiltonian canonical equations. Meanwhile,some further applications for equations such as the KdV equation, MKdV equation, the general linear Euler Lagrangeequation and the cylindric shell equations are given.     

Pade approximants and closed form solutions of the KdV and MKdV equations

The closed form solutions of the Kortewegde Vries (KdV) and modified MKdV equations which are obtainable by inverse scattering or Hirota's method are found to be Pade approximants to the formal series which results from iterating particular solutions of the linearized equation.     

Solutons of a nonisospectral and variable coefficient Korteweg-de Vries equation

A new type of KdV equation with a nonisospectral Lax pair as well as variable coefficients is introduced. Its Lax pair is shown to be invariant under the Crum transformation. This leads to a Bäcklund transformation for the KdV equation and, hence, a method for solutions via an associated nonisospectral variable coefficient MKdV equation. Three generations of solutions are given. The 1-soliton solution shares the novel phenomenology associated with the boomeron, trappon, and zoomeron of Calogero and Degasperis.     

Solitary and Periodic Waves in Coupled KdV Equations with Diferent Linear Dispersion Relations

Based on the invariant expansion method,some reasonable approximate solutions of coupled Korteweg-de Vries(KdV)equations with diferent linear dispersion relations have been obtained.These solutions contain not only bell type soliton solutions but also periodic wave solutions that expressed by Jacobi elliptic functions.The results also show that if the arbitrary constants are selected suitably,the approximate solutions may become the exact ones.     

A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV–nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions.