(2+1)维变系数KdV方程的新精确解

利用方程代换思想,对广义Riccati方程作变系数多项式展开,获得了(2 1)维变系数KdV方程的多种新精确解.相应地,亦得到近轴KdV方程的新精确解.     

变系数Fisher方程的精确解

;利用辅助方程和一种新的扩展形式解u(x,t)=,并利用符号计算系统Mathematica以构造变系数Fisher方程的精确解,包括有理函数解、三角函数解以及双曲函数解.     

KdV方程的Jacobi椭圆函数求解

把Jacobi椭圆函数展开法扩展到Jacobi椭圆正弦函数、Jacobi椭圆余弦函数和第三类Jacobi椭圆函数，并给出了KdV方程的新的周期解．并且应用这种方法得到的周期解也可以退化为KdV方程的新孤立波解．     

辅助方程法求变系数Boussinesq方程的精确解

以辅助方程法为基础,结合函数变换,借助符号计算系统Mathematica构造变系数Boussinesq方程的新的类孤子解和三角函数波解。     

两个非线性发展方程的自Bcklund变换及其精确形波解

结合截断Painlevé展式和Painlevé-Bcklund方程组的不同的解,构造了广义变系数KdV方程和（1＋1）维KdV型方程的精确形波解,并给出了这两个方程的自Bcklund变换.这个方法也可以用来构造其他非线性发展方程的精确形波解.     

变系数辅助方程法与广义Hirota-Satsuma coupled KdV系统的行波解

利用变系数辅助方程法讨论了广义Hirota-Satsuma coupled KdV方程组的精确行波解.根据齐次平衡原理又借助Maple软件计算工具获得了新的精确行波解,并且通过所得结果可以获得系统无穷多组精确行波解,丰富了该方程组的解系.     

广义变系数Kadomtsev-Petviashvili方程的孤立子解

讨论了广义变系数Kadomtsev-Petviashvili方程.首先假设存在孤波解的形式,然后利用符号计算求得其孤立子解.     

(2+1)维变系数非线性手性Schr?dinger方程的新精确解

在一些实际问题中，变系数非线性演化方程比其反常系数方程更能反映介质的非均匀性和边界的非均匀性，因此研究变系数非线性演化方程具有重要意义.对(2+1)维变系数非线性手性Schr?dinger方程进行分数阶复变换转化为常微分方程，分离实部和虚部后再分别令其为零，接着利用(G′/G²)展开法，求得了一系列带参数的精确行波通解，其中包括有理函数解、三角函数解和双曲函数解.最后当参数取特殊值时进一步得到扭结波、周期波、孤立波解等一系列新的精确解.     

一维非齐次扩散方程混合问题的形式解

利用待定函数法给出了一维非齐次扩散方程混合问题的形式解．     

二阶变系数微分方程的解法讨论

求二阶变系数线性微分方程的解,迄今为止没有一种成规的方法。本文对二阶变系数线性微分方程进行研究,从方程的自身特点出发,构造辅助函数;给出可化为常系数或可降阶的变系数二阶微分方程的条件,及在此条件下求变系数微分方程的解。     

Solutions for non-isospectral variable coefficient KdV equation

By bilinear approach we derive N-soliton-like solutions for a variable coefficient KdV equation with some x-dependent coefficients. This equation can be considered as a non-isospectral variable coefficient KdV equation. Solutions in Hirota’s form and Wronskian form are given, respectively.     

求一类偏微分方程解析新解的试探函数法

利用速探函数法,应用到KdV方程和Burgers方程和KdV—Burgers方程化为一个易于求解的代数方程,然后用待定系数法确定相应的常数,可简洁求得一类非线性偏微分方程的精确新解,此方法可望进一步推广用于求解其他非线性偏微分方程。     

Mathematical treatment of wave propagation in acoustic waveguides with n curved interfaces

There are some curved interfaces in ocean acoustic waveguides. To compute wave propagation along the range with some marching methods, a flattening of the internal interfaces and a transforming equation are needed. In this paper a local or-thogonal coordinate transform and an equation transformation are constructed to flatten interfaces and change the Helmholtz equation as a solvable form. For a waveguide with a flat top, a flat bottom and n curved interfaces, the coefficients of the trans-formed Helmholtz equation are given in a closed formulation which can be thought of as an extension of the formal work related to the equation transformation with two curved internal interfaces. In the transformed horizontally stratified waveguide, the one-way reformulation based on the Dirichlet-to-Neumann (DtN) map is then used to reduce the boundary value problem to an initial value problem. Numerical implementation of the resulting operator Riccati equation uses a large range step method to discretize the range variable and a truncated local eigenfunction expansion to approximate the operators. This method is particularly useful for solving long range wave propagation problems in slowly varying waveguides. Furthermore, the method can also be applied to wave propagation problems in acoustic waveguides associated with varied density.     

一个变系数微分方程的多种解法

通过先变形或变换,然后利用比较系数法、常数变易法、全微分法、求欧拉方程的方法、已知一个解求通解的公式等给出一个变系数微分方程的多种解法。     

Mathematical treatment of wave propagation in acoustic waveguides with n curved interfaces

There are some curved interfaces in ocean acoustic waveguides. To compute wave propagation along the range with some marching methods, a flattening of the internal interfaces and a transforming equation are needed. In this paper a local orthogonal coordinate transform and an equation transformation are constructed to flatten interfaces and change the Helmholtz equation as a solvable form. For a waveguide with a flat top, a fiat bottom and n curved interfaces, the coefficients of the transformed Helmholtz equation are given in a closed formulation which can be thought of as an extension of the formal work related to the equation transformation with two curved internal interfaces. In the transformed horizontally stratified waveguide, the one-way reformulation based on the Dirichlet-to-Neumann （DtN） map is then used to reduce the boundary value problem to an initial value problem. Numerical implementation of the resulting operator Riccati equation uses a large range step method to discretize the range variable and a truncated local eigenfunction expansion to approximate the operators. This method is particularly useful for solving long range wave propagation problems in slowly varying waveguides. Furthermore, the method can also be applied to wave propagation problems in acoustic waveguides associated with varied density.     

KdV方程的新Wronskian解

The novel Wronskian solutions of the KdV equation were obtained as limits of the soliton solutions in the Wronskian form. These solutions were verified by direct substitution to satisfy the bilinear derivative form of the KdV equadon and its Backlund transformation.     

用改进的试探函数法求解广义KdV方程

试探函数法求解非线性数学物理中一个非常著名的非线性偏微分方程-广义KdV方程，求得其一般形式的指数函数解，据此不但求得了广义KdV方程的sech^2型钟状正则孤波解，而且求得了其csch^2型奇异行波解，最后，利用一些熟知的数学关系式，又求得其若干其它显式精确解，包括三角函数型周期波解等。     

二阶变系数线性微分方程的求解

二阶线性齐次微分方程在微分方程理论中占有重要位置,但二阶变系数线性微分方程却没有一般的求解方法,给出了几种通过变量变换将二阶变系数线性微分方程化为二阶常系数的线性微分方程的充分条件.