【作者】 夏铁成；

【导师】 张鸿庆；

【作者基本信息】 大连理工大学 ， 计算数学， 2002， 博士

【摘要】 本文共分两部分，第一部分讨论微分特征列法的理论和应用问题，涉及到微分方程，抽象代数，计算机代数等重要学科。将吴方法应用到具有物理意义的线性偏微分方程上去，我们给出了Ⅱ型序，验证了张鸿庆教授八十年代给出的恰当解的概念，刻划了解的规模并给出了形式幂级数解。第二部分以构造性的变换及符号计算特别是(吴代数消元法)为工具，来研究非线性演化方程中的一些问题：精确解(如孤子解、周期解、有理解和雅可比椭圆函数解(双周期解)等)、Backlund变换、Hopf变换，dromion解及衰变结构等． 第二章介绍了求解PDEs的AC=BD模式及其在偏微分方程中的作用。首先给出了C-D对和C-D可积系统的基本理论，然后是具体研究了它们的应用。如何找到变换是这一章的一个重点内容。 第三章介绍了吴微分特征列法的最基本理论和它的应用。我们把它应用到线性偏微分方程上去得到了解的规模和形式幂级数解。因此验证了张鸿庆教授80年代的结果。 第四章讨论了双曲函数变法及其应用．推导出了双曲函数变换，利用此方法探讨了一类反应扩散方程，Brusselator反应扩散方程这些具有物理、化学、生物意义的方程的精确解(包括奇性孤波解，周期解和有理函数解)。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。在解决问题的过程中吴代数消元法是最重要的基本工具。 第五章考虑了非线性偏微分方程的雅可比椭圆函数解(双周期解)精确解的机械化算法：基于著名的sine-Gordon方程和sinh-Gordon方程，我们获得两种雅可比椭圆函数的的机械化算法。称之为雅可比椭圆函数扩展法，它是一种比sine-cosine方法和sn-cn函数法，双曲函数法更有效和简单的方法。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。 第六章研究了齐次平衡法新的应用，把它应用到WBK方程上去得到了许多新的精确解。把它应用到Boussinesq方程上去并和吴方法结合在一起获得了新的精确解。把齐次平衡法应用到BK和DLW方程上去，我们的到了许多衰变解和衰变结构。 第七章利用闫博士提出的两种扩展的Riccati方法讨论了BK，DLW，Kupershmidt等方程的精确解。我们获得了许多新解。与此同时我们也提出了一种寻找(2+1)-维或(3+1)-维NLEE类孤子解的机械化算法，应用此方法求出了(2+1)-维KD方程的许多新的类孤子解。因此我们的算法推广了范和S．A．Elwakil’s的Riccati方法。更多还原

【Abstract】 This dissertation is changed into two parts. The first part, the theory and application about Wu-Ritt differential characteristic sequence method are discussed, which involves the theories of differential equations, abstract algebra and computer algebra etc. We apply Wu-Ritt differential characteristic sequence method(abbr.Wu-Ritt method ) to linear partial differential equations which has physics significance and give the size of solutions and formal Taylor solutions. We present // formal orderings. By using // formal orderings, we make sure Zhang’s conclusion of proper solutions. The second part, with the aid of many types constructive transformation and symbolic computation(especially Wu algebraic elemination method), some topics in nonlinear evolution equation are studied, including exact solution(solitary solution, periodic solution, rational function solutions and Jacobian function solution),Backlund transformation, Cole-Hopf transformation, dromion solution and its construction etc.Charter 2 introduces AC=BD model and its application about partial differential equations. Firstly, we give a basic notion and basic theory of C-D pair and C-D integrable system and then we study their applications. How to seek for transformation u ?Cv is an important aspect in Charter 2.Chapter 3 is devoted to studying basic theory of Wu-Ritt method and its application. We apply Wu-Ritt method to linear partial differential equations and gave the size of solution arid formal Taylor solutions. Therefore we make sure Zhang’s results.Chapter 4 gives hyperbolic function transformation method and its applications. Firstly we deduce hyperbolic function transformation and then apply to a class of reaction diffusion equation and Brusselator reaction diffusion model which has physics, chemistry and biology significance. Thus we obtain many new exact and explicit solutions (including solitary wave soluiton, peoiodic wave solution and rational functions solutions) to above equations. 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.Chapter 5 considers two mechanization algorithm of Jacobian function solutions based on sine-Gordon and sinh-Gordou equations. We call above two mechanization algorithm as Jacobian function expension method. The Jacobian function expension method is an effective method than the sine-consine method and the sn-cn method. 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.In Chapter 6, we study new applications of homogenous balance method and apply the method to WBK equation and obtain many new exact solutions. We apply the method to Boussinesq equation and combine Wu algebraic elimination method, many new exact solutions are obtained. Applying the method to BK equation and DLW equation, we get mulitsolition-like solutions and many dromion solutions and their constructions.In Chapter 7, we discuss exact solutions of BK, DLW, Kupershmidt equations by using expension Riccati method which Yan and Zhang developed. Also in Chapter 7, we present a mechanization algorithm of searching for soliton-like solutions and apply the algorithm to (2+1)-dimensional KD equations and we obtain many new soliton-like solutions. Therefore we extend Fan’s and S.A.Elwakil’s algorithms.更多还原