| 1. | Chapter4 : the traveling wave method is illustrated by solving the kdv equation and sine - gordon equation
第四章运用行波法，精确求解了kdv方程和sine - gordon方程。 |
| 2. | Based on this method , the kdv equation on the initial condition is solved . and one - soliton , two - soliton and n - soliton solutions are obtained
本文运用此办法解决了kdv方程的初值问题，得到了单孤立子解、双孤立子解和n个孤立子解。 |
| 3. | That two conditions of the expansion functions should be satisfied was presented by solving the nonlinear vibrating string equation and the combined kdv equation
然后以求解非线性弦振动方程和组合kdv方程为例，指出其中的展开函数应该满足的两个条件。 |
| 4. | Finally we apply jacobian function expension method to a class of nonlinear evolution equations , rlw and a compound kdv equations and get many new jacobian function solutions and solitary wave solutions
我们分别把它应用于一类非线性演化方程， rlw和组合kdv方程上去，获得了许多雅可比椭圆函数解和其它精确解。 |
| 5. | Finally we also discuss explicit exact solutions of kdv , coupled kdv and a compound kdv - burgers equations etc . wu algebraic elimenation method is most important basic tool during the course of solving proplem
我们还研究了kdv ，耦合kdv方程及一类组合kdv - burgers方程，一类非线性演化方程精确解，这些解包括奇性孤波解，周期解和有理函数解。 |
| 6. | We took the ( 3 + l ) - dimensional nnv equation as an example , reduced it by using the travelling wave method . the translation relation between the nnv equation and one - dimensional kdv equation and that between the nnv equation and the one - dimensional mkdv equation were found
以( 3 + 1 )维nnv方程为例，采用行波法约化方程，找到了两种变换关系，把对该方程的求解分别转化为求解一维kdv方程和mkdv方程，从而得到其若干精确解。 |
| 7. | The mostly conclusion of this part is as follows , on the conditon of travelling wave , the exact solitary wave solutions to some nonlinear wave equations such as sawada - kotera equation , kaup - kupershmidt equation , the fifth order kdv equation , fisher - kolmogorov equation , on the help of the computer algebraic system ( maple ) , are explicitly established by making use of the hyperbolic function method . this part is maken up of three sections
本部分的主要结论如下，利用双曲函数展开法，在行波条件下，对sawada - kotera方程， kaup - kupershmidt方程，五阶kdv方程， fisher - kolmogorov方程，等几类非线性波动方程求解，将其孤立波表示为双曲函数的多项式，从而将非线性波方程的求解问题转化为非线性代数方程组的求解问题，并借助于计算机代数系统求解非线性代数方程组，最终获得了这些非线性波动方程的若干精确孤立波解。 |
| 8. | Chapter5 : the recently developed method of hyperbolic tangent function expansion is extended and new function transformation is applied to construct some new solitary solutions of kdv equation and klein - gordon equation and the jacobi elliptic function expansion method , which is advanced in 2001 , and the extended method of doubly jacobi function expansion are used to construct the exact solutions of a kind of nonlinear evolution equations
第五章对近年来发展起来的双曲正切函数展开法加以改进，采用新的变换函数，得到了kdv方程和非线性klein - gordon方程的一些新的孤立波解。其次，分别采用2001年提出的jacobi椭圆函数展开法和本文由此扩展而来的双椭圆函数展开法，求解了一大类非线性发展方程，得到了一系列新的周期解。 |
| 9. | Chapter1 : the developement of the theory of soliton is presented . the famous kdv equation is introduced in detail , which plays an important role in the theory of nonlinear equations . and the problem of the interaction between solitons is also studied , which shows that the traveling solitons keep steady after collision
第一章介绍孤立子理论发展概况，详细推导了在非线性方程理论研究中具有重要意义的非线性波动方程kdv方程，并且研究了孤立子相互作用问题，分析表明孤立了碰撞以后形状保持稳定。 |
| 10. | ( 1 ) based on two types of riccati equations , two kinds of new methods are proposed to obtain solutions of nonlinear differential equations . twelve families of exact solutions of wbk equation are found by using one of two methods ; ( 2 ) the homogeneous balance method is improved cind investigated to ( 2 + l ) - dimensional broer - kaup equation such that many families of new solutions are derived . ( 4 ) based on the isospectral lax pair of riccati form for generalized kdv equation with the force term , new darboux transformation and solitary - like wave solutions and rational solutions are obtained ; ( 4 ) by constructing darboux transformation and the superposition formula of generalized variable coefficients kdv equation with the force term , new single solitary - like wave solutions , double solitary - like wave solutions and rational solutions are found for ( 2 + l ) - dimensional generalized kp equation
第二章和第三章考虑非线性偏微分方程的精确解的构造：首先给出了c - d对和c - d可积系统的基本理论，然后在第三章中具体研究了它们的应用： ( 1 )基于两种riccati方程，提出了两种新的求解非线性微分方程更多解的方法，利用其中的一种方法，得到了wbk方程的12组精确解； ( 2 )对齐次子衡法进行改进，以致于获得了( 2 + 1 ) -维broer - kaup方程的很多新解； ( 3 )基于带有外力项的广义kdv方程的riccati形式的非等谱lax对，提出了该方程的一个新的darboux变换，利用该变换，得到了新的类孤波解和有理解； ( 4 )通过构造了带有外力项的变系数kdv方程的darboux变换及叠加原理，获得( 2 + 1 ) -维广义kp方程的新的类单孤波解、双类孤波解和有理解。 |
