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).  

KdV-Burgers方程的孤波解

对双曲函数法进行了深入探讨,推广了该方法的某些使用条件,借助计算机代数系统Mathe matica,进一步获得了KdV-Burgers方程的两组扭状孤波解.  

修正Jacobi椭圆函数展开法及其应用

对Jacobi椭圆函数展开法进行了扩展, 且应用修正过的方法获得了若干非线性波动方程的更多的准确周期解, 补充了前面研究所得的结果.  

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.  

非均匀尘埃等离子体中孤子的传播

运用约化摄动法研究了非均匀尘埃等离子体中孤子的传播情况. 在低阶近似下, 对于小的、但有限振幅的长波振动, 当分界面不连续变化时,孤子在不连续点的反射波与透射波均可由 KdV 方程来描述, 并给出了低阶近似情况下, 对于小的、但有限振幅的长波振动, 当入射波为单孤子时, 反射孤子与透射孤子的个数及其大小;当分界面是有限长度并连续变化时,对于小的、但有限振幅的长波振动, 尘埃声孤波由KdV型方程来描述,并由此给出了准孤子振幅、传播速度等参量在传播过程中的变化.  

立方非线性Schr(o)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).  

Nonlinear Wave in a Disc-Shaped Bose-Einstein Condensate

We discuss the possible nonlinear waves of atomic matter wave in a Bose-Einstein condensate. One and two of two-dimensional （2D） dark solitons in the Bose-Einstein condensed system are investigated. A rich dynamics is studied for the interactions between two solitons. The interaction profiles of two solitons are greatly different if the angle between them are different. If the angle is small enough, the maximum amplitude during the interaction between two solitons is even less than that of a single soliton. However, if the angle is large enough, the maximum amplitude of two solitons can gradually attend to the sum of two soliton amplitudes.  

An exact solution of Fisher equation and its stability

In this paper, the Fisher equation is analysed. One of its travelling wave solution is obtained by comparing it with KdV-Burgers （KdVB） equation. Its amplitude, width and speed are investigated. The instability for the higher order disturbances to the solution of the Fisher equation is also studied.  

柱和球尘埃离子声孤波的绝热近似解

通过运用等价粒子理论，得到了尘埃声孤波中的KdV类型方程（包括KdV方程，柱KdV方程和球KdV方程）的绝热近似解。这种方法也可以运用到其它的非线性演化方程。  

Analytical solutions for the spin-1 Bose-Einstein condensate in a harmonic trap

The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.