【作者】 徐桂琼；

【导师】 李志斌；

【作者基本信息】 华东师范大学 ， 系统分析与集成， 2004， 博士

【摘要】 非线性演化方程是描述物理现象的一类重要数学模型，也是非线性物理特别是孤立子理论最前沿的研究课题之一。非线性演化方程精确解和可积性的研究有助于弄清物质在非线性作用下的运动规律，对相应物理现象的科学解释和工程应用将起到重要作用。在非线性演化方程的研究中，寻找方程的行波解、构造多孤子解、Painlevé可积性质的检验等经常遇到复杂的符号计算和推理，有的是人力难以完成的，因此妨碍了这些问题的深入剖析。近年来，符号计算的蓬勃发展，极大地推动了非线性演化方程的研究。非线性演化方程的研究成果不断涌现，尤其是新的求解方法层出不穷。本文以非线性演化方程为研究对象，借助于符号计算这一有效研究工具，研究了多种直接代数方法在非线性演化方程精确求解中的应用、Painlevé分析及其应用，探讨了几种直接代数方法与Painlevé可积性质之间的内在联系。主要工作如下： 第一部分研究非线性演化方程的精确求解。分别从三个方面进行研究： 研究了构造非线性演化方程孤立波解的基础性方法——混合指数方法，改进了混合指数方法的关键步骤——行波约化后常微分方程及递推关系式的求解。将传统实指数方法推广到复指数情形，从而可以获得正则孤波解、奇异孤波解及周期解在内的诸多形式的行波解。 在Riccati方法、形变映射方法、“统一代数”方法的基础上，给出了构造非线性演化方程多种行波解的广义形变映射法。该方法的基本思想是利用“秩”对行波约化后的常微分方程进行分类，对同秩类型和异秩类型的方程借助于不同的一阶可解方程，从而将非线性演化方程行波解的计算问题转化为非线性代数方程组的求解问题。利用吴文俊消元法求解非线性代数方程组，即可得到非线性演化方程的多类行波解。以耦合mKdV方程组、耦合Drinfel’d-Sokolov-Wilson方程组、变形Boussinesq方程组等为例，系统地获得了包括指数解、多项式解、有理解、三角函数解、Jacobi椭圆函数解、Weierstrass椭圆函数解、孤立波解及广义孤立波解、组合形式三角函数及孤立波解在内的多种形式的行波解。由于该方法是构造性和算法化的，可以在计算机代数系统上完成解的自动推导。 采用分步确定拟解的原则，对齐次平衡法求非线性演化方程多孤子解的关键步骤作了进一步改进。以广义Boussinesq方程和bKK方程为应用实例，说明使用该方法可有效避免“中间表达式膨胀”的问题，除获得标准Hirota形式的孤子解外，还能得到其他形式的孤子解。本文获得了这两个非线性演化方程的具有双向传播特点的孤立波解和孤立子解。 第二部分研究非线性演化方程的Painlevé可积性的检验和Painlevé分析的若干应用。主要从两个方面进行研究： 由于非线性系统的Painlevé性质与可积性之间有着十分密切的联系，因此判定一个非线性系统是否具有Painlevé性质就具有非常重要的意义。本文分析了非线性演化方程Painlevé性质的几种检验方法，并利用Kruskal方法得到了广义Hirota-Satsuma耦合方程组具有Painlevé可积性质的必要条件。WTC方法和Kruskal方法是检验非线性演化方程Painlevé性质的两种重要方法，这两种方法各有所长，本文将二者结合起来并给出了检验Painlevé性质的WTC-Kruskal算法。使用WTC-Kruskal算法，不仅可以快速判定非线性演化方程的Painlevé可积性，也为寻找新的Painlevé可积系统提供了重要途径。 研究了基于Painlevé性质的若干截断展开方法在非线性演化方程可积性质及精确解研究中的若干应用。讨论了标准截断展开方法在构造不可积系统精确解、可积系统的自B(?)cklund变换、多重孤波解和多孤子解中的应用.其次，给出了高阶截断展开法在构造非线性演化方程新型精确解中的应用，并分析了高阶截断展开法在构造非线性可积系统Lax对、Darboux变换的应用实例.最后将近年来发展起来的构造非线性演化方程行波解的几种直接代数方法统一于Painle说分析的研究框架之中. 第三部分研究符号计算在广义形变映射法和Painlev乙可积性证明中的应用.主要包括: 广义形变映射法将行波解的求解间题转化为非线性代数方程组的计算问题，推演过程往往涉及到非常繁琐的计算.本文在计算机符号系统Maple上开发了一个基于吴文俊消元法的行波解自动求解软件包NETs.该软件包可以自动输出非线性演化方程及方程组的多类行波解.NETS软件包对方程或方程组的维数没有限制，适用于多项式类型的非线性演化方程.除适用于非线性偏微分方程外，NETS还适用于非线性常微分方程.某些特殊类型的非线性演化方程，经过适当的函数变换转换为多项式类型的非线性常微分方程后，可以借助NETS实现行波解的自动推导. 基于wTC一Kruskal算法，本文在计算机符号系统Maple上开发了wkPtest软件包.该软件包可快速完成非线性偏微分方程及方程组Paiulev‘性质的自动检验.同时，当给定方程不能通过Palnlev乙检验时，软件包将返回参数满足的约束条件.此外，软件包还能输出方程的Painlev‘截断展开式.wkPtest软件包适用范围较为广泛，对方程或方程组的维数没有限制，不仅适用于常系数方程或方程组，也适用于变系数方程或方程组以及一?更多还原

【Abstract】 Nonlinear evolution equation is an important mathematical model for describing physical phenomenon and an important field in the contempary study of nonlinear physics, especially in the study of soliton theory. The research on the explicit solution and integrability are helpful in clarifying the movement of matter under the nonlinear interactiveties and play an important role in scientifically explaining of the corresponding physical phenomenon and engineering application. Many research topics, such as searching for exact explicit solutions, multi-soliton solution, the Painleve test et al., often involve a large amount of tedious algebra auxiliary reasoning or calculations which can become unmanageable in practice. In recent years, the development of symbolic computation accelerates the research of nonlinear evolution equation greatly. Many new methods for constructing exact solutions of nonlinear evolution equations are proposed. This dissertation mainly stduies some aspects of nonlinear evolution equations with the aid symbolic computation, which include searching exact explicit solutions of nonlinear evolution equations by means of several direct algebraic methods proposed in recent years, the Painleve analysis and its application, the interraltions between the algebraic methods with the Painleve analysis. This dissertation consists of the following three parts.Part I is devoted to study the explicit solutions for nonlinear evolution equations by some algebraic methods proposed in recent years.Mixing exponential method proposed by Hereman for finding the solitary wave solutions to a nonlinear evolution equation is developed and perfected. Correspondingly, an extended mixing exponential method is obtained by expressing the solutions as an infite series of the real or complex exponential solutions of the underlying linear equations and improving the solving of recursion relations. The effectiveness of the extended approach is demonstrated by application to some nonlinear evolution equations with physical interest. Not only are steady solitary wave solutions recovered, but also the diverging and the periodic solutions are obtained.Based on Riccati method, deformation mapping method and unified algebraic method, a generalized deformation mapping method is presented for finding travelling wave solutions to nonlinear evolution equations. According to "rank", the nonlinear evolution equations are classified into two kinds of equations. The essence of this method is to take full advantage of two different solvable first-order ordinary differential equations, and convert the problem of finding travelling wave solutions for nonlinear PDE to the problem of solving nonlinear algebraic equations. The nonlinear system is solved by using Wu elimination and a series of travelling wave solutions are then obtained. Several illustrative equations such as coupled mKdV equations, coupled Drinfel’d-Sokolov-Wilson equations, variant Boussinesq sytem etc. are investigated by this metho. A series of travelling wave solutions are obtained in a systematic way, which covered exponential function solutions, polynomial function solutions, rational function solutions, trigonometric function solutions, Jacobi elliptic function solutions, Weierstrass elliptic function solutions, solitary wave solution, combined-form solitary wave solutions and so on. The generalized deformation mapping method is completely algorithmic, so it can be implemented in computer algebra.The homogeneous balance method for constructing solitary wave solutions and soliton solu-tions is further developed on obtaining quasi-solution by step-by-step principle. The main advantage of the extended approach is to avoid the problem of "intermediate expression swell". The effectiveness of the method is demonstrated by application to the generalized Boussinesq equation and the bidirectional Kaup-Kupershmidt equation. The one-soliton, two-soliton and three-soliton solutions with multiple collisions are derived for these two equations with the assistance of Maple. Both更多还原