若干非线性发展方程的精确解

【作者】 何煜；

【导师】 屈长征；

【作者基本信息】 西北大学 ， 应用数学， 2011， 硕士

【摘要】 非线性发展方程精确解的求解一直受到数学家和物理学家的热切关注.目前虽然已经建立和发展了不少非线性发展方程行之有效的求解方法,但由于非线性理论的复杂性,对于大量的非线性发展方程仍然无法求得其精确解.因此,继续寻找一些有效可行的求解方法是科学研究中一项十分重要和极具价值的工作.本文对首次积分法的思想方法及后人对此方法的改进做出了介绍,并对一些有重要物理意义的非线性发展方程进行了求解,所得结果不仅囊括了若干已知的精确解,而且还得到了许多新的精确解.丰富及完善了这些非线性发展方程精确解的研究.全文分为三章.第一章为绪论,介绍了非线性发展方程的基本概况和非线性发展方程精确解的研究现状,并扼要介绍了本文的主要工作.第二章系统介绍了首次积分法及后人对此方法的改进及具体的应用操作过程.第三章运用首次积分法对Modified Improved Boussinesq方程、(2+1)维色散长波方程组、(2+1)维Konopelchenko-Dubrovsky方程组进行了详细的求解,并获得了方程(组)新的精确解.最后我们对本文的工作进行了总结,提出了一些自己有待研究的问题并对今后研究方向作出了展望.更多还原

【Abstract】 Searching for exact solutions of nonlinear evolution equation has long been a popular research subject for mathematicians and physicists.At present, there has been established and developed a lot of effective method to solve nonlinear evolution equations, but a large number of nonlinear evolution equations are still unable to solve for complexity of nonlinear equations themselves.As a con-sequence,it is a very important research and valuable work to go on searching for some effective and feasible method to solve nonlinear evolution equation.In this paper, the first integral method of thinking and successor to make improvements to this method is introduced.By using this approach,a lot of exact solutions for some nonlinear evolution equations which have important physical significance are easily presented. The results include not only a number of known exact solutions, but also a number of new exact solutions.This enriched and improved the content of exact solutions of nonlinear evolution equations.This paper is composed of four chapters:In the fist chapter, we introduce the basic profiles of nonlinear evolution equations and the research status of exact solutions for nonlinear evolution equa-tions.The main work of this article is introduced.In the second chapter,we elaborate the first integral method and its appli-cation procedure in detail.In the third chapter, we make use of first integral method to obtain a lot of new exact solutions of Modified Improved Boussinesq equation, (2+1)-dimensional dispersive long wave equations and (2+1)-dimensional Konopelchenko-Dubrovsky equations.In the end, we summarize the work of this paper and look into the distance of research in future as well.更多还原