Main object of the first chapter is to investigate adaptive mesh re - distribution methods . successful implementation of the adaptive strategy can increase the accuracy of the numerical approximations and also decrease the computational cost 移动网格方法，作为自适应方法的一种，主要是为了解决发展方程的计算问题而设计的方法，本文第一部分主要研究的是一种自适应网格重新分布方法。
In this paper we study the existence uniquness and the asymptotic behavior of global solutions and the nonexistence of the global solutions for the initial boundary value problems , cauchy problem and abstract cauchy problem to some nonlinear evolution equations of higer order 本文讨论几类非线性高阶发展方程(组)初边值问题、 cauchy问题和抽象初值问题整体解的存在唯一性、解的渐近性以及整体解的不存在性，主要结果有以下五部分内容
( 2 ) by the jacobi elliptic function expansion method and computer symbolic systems ( maple , matlab ) , many exact periodic solutions of nonlinear wave equations can be obtained and these solutions can degenerate solitary wave solutions , shock wave solutions , trigonometric function solutions ( 2 )利用jacobi椭圆函数表示法和数学计算软件( maple ， matlab等) ，可以得到一类常系数非线性发展方程的双周期解，这些解能退化成孤立子解，冲击波解，三角函数解。
With the development of science and technology , there are many nonlinear problems in natural and social areas , which arouses much concern . these problems are usually characterized by nonlinear evolution partial differential equations ( nlepdes ) . so how to construct exact solutions of the associated nonlinear equations plays an important role in understanding the nonlinear problems 随着科学技术的发展，在自然科学和社会科学领域中广泛存在着的非线性问题，越来越引起人们的关注，而且许多非线性问题的研究最终可归结为非线性发展方程来描述，因而如何得到它们的精确解对研究相关的非线性问题非常重要。
First according to the fact that tangential components of the evolution do not affect the geometric shape of the evolving curves , we introduce the evolution equation of geometric quantities for the general planar curves . then we describe the work of gage - hamilton briefly . last we consider a special curvature flow of curve which evolves with speed function of the principal curvatures along the inner norm and show that convexity of the curve is kept and its length and area are contracted if the initial closed curve is smooth and convex . so the final shape of the curve will be a point in finite time 首先根据曲线在切向分量上发展是不影响曲线的发展形状，我们引入了曲线的一些几何变量的发展方程；其次我们简要地回顾gage - hamilton研究曲线发展的一般步骤；最后我们考虑沿曲线的内法线以曲率的函数为发展速度的一类特殊的曲线族，证明了在初始曲线为凸的闭平面简单曲线条件下，曲线将保持凸的，并且它的面积和周长将同时收缩，并在有限时间内成为一个点。
In this paper , we consider the global smooth solutions and long time be - haviors for some nonlinear evolution equations , such as kdv equation , bbm equation , gbbm equation , kdv - burgers equation , coupled generalized nonlin - ear wave equation . by using a priori estimates , the existence of global smooth solution , the existence of global attractors and its fractal dimensions for this sys - tems are obtained . this paper is organized in six chapters 本文考察了kdv 、 bbm 、 gbbm 、 kdv - burgers 、广义耦合的非线性波动方程组等非线性发展方程整体光滑解及其渐近行为，利用先验估计，对一类广义kdv方程组及耦合的波动方程组的周期初值问题、 cauchy问题、初边值问题进行了讨论，研究了整体吸引子的存在性及其分形维数有限性估计。
And the newly developed method of lame function method in solving the multi - order approximate equations of nonlinear evolution equations is also discussed . and following equations are solved in this method : nonlinear schr dinger , bbm equation , zakharov equation , kp equation , boussinesq equation and cubic nonlinear schr dinger 最后介绍在椭圆函数展开法基础上发展而来的，利用lam函数求解非线性发展方程多级近似解的方法，并且求解了非线性schrdinger方程，非线性bbm方程， zakharov方程， kp方程， boussinesq方程和立方非线性schrdinger方程等方程。
In the part of discussion , the suitability of the jacobi elliptic function expansion method is also studied by proposing the " rank " . and we firstly point out that when the " ranks " of every term of the nonlinear evolution equation are simultaneously even or odd , the method can be used to solve the equation 为了讨论了jacobi椭圆函数展开法的适用性问题，我们最先引进“秩”的概念，指出只要非线性发展方程的各项的“秩”满足相同的奇偶性，就可以用这种展开法求解。
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。
Abstract : the stability of the nonlinear comput ation is important in numerical weather prediction . in this paper , the concept , phe n omena and mechanism of nonlinear computational instability of the nonlinear evol ution equations in numerical weather prediction are discussed . an effective schem e , the square conservation shceme , is presented 文摘:简述了数值天气预报问题中非线性发展方程的非线性计算不稳定性的含义、数值计算现象以及计算不稳定产生的机理，给出了克服非线性计算不稳定的一个有效方法? ?平方守恒格式。
