基于李对称分析的偏微分方程精确解的研究
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摘要
偏微分方程又称数学物理方程,它来源于物理学、力学等自然科学及工程技术中所提出并建立的数学模型。早期的偏微分方程有根据牛顿引力理论推导出的描述引力势的拉普拉斯(Laplace)方程和泊松(Poisson)方程,还有描述波的传播的波动方程(wave equation),描述传热和扩散现象的热传导方程(heat equation)等,这些都是古典的偏微分方程。这些方程在偏微分方程理论的发展中发挥了重要的作用,时至今日,它们仍然是偏微分方程的基础和必学内容之一。自19世纪开始,随着工业革命的兴起和科学技术的发展,相继出现了大量新的偏微分方程,其中最基本的有描述电磁场变化的麦克斯韦方程(组),描述微观粒子的薛定谔方程,以及爱因斯坦方程、杨-米尔斯方程、反应扩散方程等等。随着现代科学和技术的进步,还将会不断涌现出新的越来越多的偏微分方程,尤其是非线性的偏微分方程或方程组。其中,非线性波方程是描述自然现象的一类重要数学模型,也是非线性数学物理特别是孤立子理论最前沿的研究课题之一。通过对非线性波方程的求解和定性分析的研究,有助于人们弄清系统在非线性作用下的运动变化规律,合理解释相关的自然现象,更加深刻地描述系统的本质特征,极大地推动相关学科如物理学、力学、应用数学以及工程技术的发展。
本文以李(S.Lie)对称分析为基础和工具,综合运用动力系统的分支理论与方法、潘勒维尔(Painleve)分析、幂级数法(含推广的幂级数法)、待定系数法以及一些特殊的技巧与方法,研究偏微分方程的精确解及其相关的方程与解的性质。具体而言,即首先运用李对称分析得到方程的向量场或对称,然后利用相似约化将所研究的(非线性)偏微分方程化为常微分方程。这一步对方程而言可以说实现了实质性的转化,即把一个复杂的偏微分方程,包括各种非线性的、变系数的偏微分方程转化为一个常微分方程。接下来的工作就是研究这个常微分方程的解,求出了常微分方程的解,也就相应地得到了偏微分方程的解。这就是利用对称分析研究偏微分方程精确解的基本思路。当然,对称分析的作用远不止此,它与系统的可积性的研究还有着密切的关系,对称是系统本质属性的一种描述和刻画,它在偏微分方程与可积系统的研究中有着重要的意义与作用。这些我们将在研究偏微分方程精确解的同时一并加以介绍。至于如何研究约化得到的常微分方程,则主要涉及常微分方程与动力系统的理论与方法、幂级数法以及一些特殊的技巧与方法。
本文的主要内容如下：
第一章绪论。本章介绍了非线性科学的主要内容以及发展现状,综述了偏微分方程,尤其是非线性波方程的发展历史、研究现状、主要研究方法以及取得的主要成果。其中重点介绍了偏微分方程研究的主要方法,特别是对称分析在研究偏微分方程中的意义与作用。概括而言,这些方法各有特点,也都有各自的适用范围,都在特定的时期、特定的条件和各自的范围内发挥了应有的作用。有的方法可以说长盛不衰,历久弥新,至今还有强大的生命力,在偏微分方程的研究中仍然发挥着重要的作用。当然,任何一种方法都不是万能的,不会也不可能指望用一种方法解决所有的问题。本章的出发点是对各种主要的方法加以总结回顾,目的不是评判哪种方法的优劣,而是通过比较和总结,更好地继承和发扬其中蕴含的优秀的思想方法,从过去经典的思想与方法中汲取营养,更好地面向未来,进一步更深入地开展对现代偏微分方程及相关非线性科学的研究。
第二章理论准备。在这一章,列举了本文所涉及的一些相关知识,如李群与李代数、对称与向量场、向量场的延拓、Painleve分析简介、动力系统的分支理论与方法以及雅可比(Jacobi)椭圆函数等。限于篇幅,有些内容只列出主要概念与结论,详细内容可查阅后面的相关参考文献,此处不展开叙述。单列本章的目的是考虑到李群与对称分析的相关理论与知识比较多,通过本章,对有关的理论知识有所了解,便于后面的具体运用。
第三章基于李对称分析,研究了一般的Burgers'方程。该方程是一个既有非线性项又有二阶偏导项的非线性波方程,在理论和实践中有广泛的应用价值。它在一定条件下存在不同类型的孤波解,如冲击(震荡)波、稀疏波等。在流体力学、空气动力学的许多波动问题的研究中都要用到这个方程。例如在流体力学模型方程中,有线性Burgers'方程ut+aux=μuxx和非线性Burgers'方程ut+[f(u)]x=μuxx。当f(u)=1/2u2时,后者即为ut+uux=μuxx。在一定的初、边值条件下,可以得到这两类Burgers'方程的精确解,从而了解系统相应的流体力学性质。另外,Burgers'方程和许多重要的数学物理方程有着密切的联系,在非线性科学、流体力学以及工程技术中起着重要的基础性作用。在对称分析的基础上,首先求出了方程的群不变解以及任意次的迭代解。然后,利用对称约化将原方程化为各种形式的常微分方程,进而求出方程的精确解。其中应用了幂级数法(Power series method),得到了非线性、非自治的常微分方程严格的幂级数解,从而也就得到了相应的Burgers'方程的精确解,其中包含了不少新的显式精确解。
第四章研究推广的mKdV方程,众所周知,KdV方程是非常著名的浅水波方程,它起源于对水波问题的研究,KdV型方程可以描述各种浅水波的运动,在流体力学中有着广泛的应用。特别地,对于修正的KdV型方程,最近的研究发现可用于描述宇宙环境中超新星周围以及土星环的尘埃离子的波动规律,对于天体力学和大气物理的研究有着重要的意义。首先,通过对称分析得到了它的向量场。然后,由一般到特殊地得到了一些特殊而经典的KdV、mKdV方程的向量场。接下来,通过对称约化将推广的mKdV方程化为常微分方程,为下一步求解作准备。本章的一个亮点是运用了动力系统的分支理论与方法,详细全面地得到了推广的mKdV方程的显式精确解,包括幂级数解,同时还研究了系统的动力学性质。
第五章研究了一类短脉冲方程的精确解。短脉冲方程也是一类非常重要的非线性波方程,可以描述一些比较特殊的波。深入研究这类方程及其各种孤波解,对于了解一些特殊的波动问题具有重要意义。同时,该方程是一个重要的非线性数学物理方程,它在工程技术以及物理学、力学的许多领域都有重要应用。此方程不同于一般的非线性演化型方程,而是一个混合型的偏微分方程,这给对称分析带来了一定的困难。本章分别运用延拓法与待定系数法,得到了该方程的所有对称。其次,本章的另一特色是在运用动力系统的分支理论与方法研究方程的精确解时,引入了参数表示法,从而圆满地解决了解的显式表示问题。本章获得的这类短脉冲方程的精确解,都是用通常的方法难以得到的。
第六章研究了一类变系数债券方程。变系数偏微分方程最初主要来源于数学物理问题及大量的工程技术问题,但是,随着社会的进步和现代科学技术的不断发展,在各种经济社会领域、生物化学与环保领域、通讯信息与金融证券等领域,由于实际的需要也提出了越来越多的偏微分方程,这些方程一般形式复杂,且常常是变系数的。本章研究的变系数方程在金融数学与金融工程中经常用到,尤其是在期权定价问题的研究中,这类偏微分方程发挥着日益重要的作用。偏微分方程理论与现代经济、金融研究相结合,正成为一种重要的发展趋势。首先,对两个具体的变系数债券方程进行了对称分析,分别得出了它们的向量场。然后,又分别求出了它们的单参数群与群不变解。第三,利用相似变换分别将它们约化为常微分方程。第四,进一步求出它们的精确解。本章在内容上与前几章的主要不同之处在于,一是对称分析,由于所研究的方程是变系数的,因此,对称分析要比常系数方程复杂得多。二是在求精确解时除了幂级数法之外,还用了待定系数法等一些特殊方法,从而得到了方程的显式精确解,收到了较好的效果。三是在本章最后,我们还就一般形式的变系数债券方程进行了讨论,得出了它的对称及相应的精确解。
第七章研究了三个非线性演化方程。这类方程在非线性科学与工程技术中有着重要的意义与作用,是许多波动问题和力学问题的重要理论模型,在生物数学等领域也有着重要的应用。首先运用Painleve分析得到了它们的Painleve性质,以及相应的Backlund变换、截断展开式等。然后再通过对称分析,分别得到了它们的对称,并通过比较分析了Painleve分析与对称分析的异同。接着研究它们的精确解,除了基于对称分析的精确解,我们还得到了方程的基于Painleve截断展开的精确解。这些解的获得,是单独用任何一种方法所不可能得到的,这也说明了二者结合的意义和作用。另外,通过本章的研究可以发现,对于有些即使是不可积的方程,我们仍然可以利用对称分析与Painleve分析研究它们的精确解。我们知道,在可积系统的研究中,Painleve分析的主要作用是判断系统的可积性,但通过本章可以发现它还可以用于方程求解的研究。对称分析更是如此,无论是否可积,都可以通过对称分析研究方程的精确解。
总之,本文研究的对象是偏微分方程,包括各种非线性的、变系数的方程。主要目的是求出方程的解,尤其是显式的精确解。所以,本文所采用的方法与工具与一般孤子与可积系统的研究有所不同,结果也不一样,可以说各有侧重。限于论文的主题,尽管系统的对称与可积性如守恒律(CL)、Backlund变换等有着密切的联系,但对系统的可积性不作过多的讨论,目的是使论文主题更突出。另外,这些方程都是重要的数学物理方程,深入研究这些方程的解及其相关性质,如Painleve性质、可积性以及各种形式的解,尤其是各种显式精确解,对于了解系统所描述的具体问题的性质与规律,有着重要的意义与作用。
最后,在总结与展望中,首先概述了本文所获得的主要研究成果；然后,总结归纳了本文的主要创新点；最后,提出了围绕偏微分方程精确解的研究有待于进一步研究与思考的方向和问题。
Partial differential equations (PDEs), also called mathematical physics equations, are derived from the models of Physics, Mechanics and Engineering, etc. The earlier PDEs, such as Laplace equation and Poisson equation are based on the Newton's theory, wave equation, heat equation, and so on. And these equations are all classical PDEs. From the 19th century, a lot of new PDEs are derived, including the famous Maxwell system, Schrodinger equation, Einstein equation, Yang-Mills equation, and reaction-diffusion equation, etc. More and more PDEs, especially the nonlinear equations and systems will be derived owing to the progress of the modern science and technology. In addition, the nonlinear wave equations are important mathematical models for describing natural phe-nomena and one of the forefront topics in the studies of nonlinear mathematical physics, especially in the studies of soliton theory. The research on finding explicit and exact solu-tions of nonlinear wave equations and on analyzing the qualitative behavior of solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions, explain the corresponding natural phenomena reasonably, describe the essential properties of the nonlinear systems more deeply, and promote greatly the development of engineering technology and related subjects such as physics, mechanics and applied mathematics.
In this paper, the exact and explicit solutions, the dynamical behavior and the inte-grable properties for PDEs are investigated from the viewpoint of bifurcation theory of dynamical systems, Painleve analysis, (generalized) power series method, method of un-determined coefficient and a lot of special techniques, based on the Lie symmetry analysis. In detail, firstly, we solve the vector field or symmetries for a PDE with the Lie symmetry analysis method. Then, we reduce the PDE to the ordinary differential equations (ODEs) by the similarity transformations, it is a significant progress for dealing with the exact solutions of a PDE. That is, we transform a complicated PDE, including the nonlinear equation, variable coefficient equation, etc., to a ODE. Thirdly, we deal with the ODEs with the various method, such as the bifurcation theory of dynamical systems, Painleve analysis, (generalized) power series method, method of undetermined coefficient and a lot of special techniques. This is the outline for studying the exact solutions for PDEs. As we all know, by employing the Lie symmetry analysis, we can also consider the the integrable property for PDEs. But this is not the theme for this thesis, so we will not investigate it in detail.
The major works of this thesis are as follows.
Chapter 1 is the introduction. In this chapter, the historical background, research developments, main methods and achievements of partial differential equations especially the nonlinear wave equations are summarized. The discovery, corresponding research approaches and recent advance of Lie symmetry analysis and Painleve analysis methods are introduced. The relationship between partial differential equations and dynamical behavior, Painleve properties along with the study on exact solutions of partial differential equations by using the Lie symmetry analysis method are presented.
Chapter 2 is the preparation of theory. In this chapter, some preliminary knowledge of dynamical systems, theory of Lie groups and Lie algebras, prolongation of vector fields and the other basic mathematical theory and main results are introduced. This chapter is the preliminary knowledge of the thesis. For the sake of succinctness, we outline the summaries only. If it is necessary to learn some contents in detail, please consult the references.
In Chapter 3, the Lie symmetry analysis method is performed for the general Burg-ers'equation. This equation is a nonlinear wave equation, and it is of great importance in both theory and applications. The equation has various solitary wave solutions, such as the shock wave solutions, sparse wave solutions, etc., so it is important to study the wave problems in fluid dynamics and aerodynamics. For example, in the model equations of fluid dynamics, we have the linear Burgers' equation ut+aux=μuxx and the nonlinear Burgers'equation ut+[f(u)]x=μuxx, respectively. In particular, if f(u)=1/2u2,then the latter is ut+uux=μuxx.Under the certain conditions, we can get the exact solutions for the two equations. Thus, the fluid dynamical property can be comprehended. More-over, the Burgers'equation can be transformed into other important mathematical physics equations such as the heat equation, so it plays a significant role in the nonlinear science, fluid dynamics and engineering.
Based on the Lie symmetry analysis, we get the group-invariant solutions, iterated solutions. Then, by using the similarity reductions, we transform the general Burgers' equation to ordinary differential equations (ODEs), some of them are nonlinear and non- autonomous quations. Next, we deal with these ODEs by using the power series method and some special techniques, the exact solutions for the ODEs are obtained, so the solu-tions of the general Burgers'equation are presented simultaneously. Some new solutions are obtained for the first time in this chapter.
In Chapter 4, we investigate the extended mKdV equation. As is well known, the KdV equation is a famous shallow water wave equation, and it comes of the studies on the water wave problems. The KdV type equations can depict the various shallow water wave moment, and they play an important role in fluid dynamics. The modified KdV (mKdV) equation, on the other hand, has recently been discovered, e.g., to model the dust-ion-acoustic waves in such cosmic environments as those in the supernova shells and Saturn's F-ring, etc.
By using Lie symmetry analysis method, the vector field for the extended mKdV equation is obtained, and the vector fields of the several special classical KdV-type equa-tions are presented simultaneously. Then, by using the method of dynamical systems for the extended mKdV equation, all the exact solutions based on the Lie group method will be given. Especially, the bifurcations and traveling wave solutions are obtained. To guar-antee the existence of the above solutions, all parameter conditions are determined. Fur-thermore, the exact analytic solutions are considered by using the power series method. Such solutions for the equation are important in both applications and the theory of nonlin-ear science. Note that the vector fields of the several special classical KdV-type equations obtained in this way are a part of the vector fields rather than its complete vector fields, respectively. This reflect the intricacy of the Lie symmetry analysis. One of the highlight of the chapter is the method of dynamical systems. By employing this method, we discuss the dynamical behavior and get the all traveling wave solutions for the equation.
In Chapter 5, the Lie symmetry analysis and the generalized symmetry method are performed for a short pulse equation (SPE). The SPEs are nonlinear wave equations also, it can depict some peculiar waves. It is of great importance in comprehending some special wave problems that the SPEs and its solitary wave solutions are investigated in detail. At the same time, the SPEs are important mathematical physics equations, it play a significant role in engineering and mechanics.
Note that this equation is not a common evolution equation, but a mixed one. This brings some difficulty for the Lie symmetry analysis method. The symmetries for this equation are given. For the traveling wave solutions, the exact parametric representations are investigated. To guarantee the existence of the above solutions, all parameter condi-tions are determined. Furthermore, the exact analytic solutions are obtained by using the power series method. One of the highlight of the chapter is that we use the method of undetermined coefficient to solve the symmetries for the equation. Then, for getting the exact explicit solutions, the method of parameters are employed. It is not obtained by the general method for these solutions obtained in this chapter.
In Chapter 6, we use Lie symmetry group methods to study a pair of bond pricing equations. The variable-coefficient PDEs are derived from the mathematical physics and engineering problems at first. Now, many PDEs are derived from society, biochemistry, environmental protection, information science, communication, finance and securities, etc., along with the progress of society and the development of science and technology. Moreover, these PDEs are complicated and variable-coefficient usually. The variable-coefficient PDEs in this chapter are of importance in financial mathematics and financial engineering, and it play an important role in bond price as well.
The symmetries and similarity transformations for the two equations are provided, and all exact solutions are obtained. The general caseν≥2 are considered simultane-ously. We show that when the reduced equations are obtained, the generalized power se-ries method and special techniques can be used to find the exact solutions.When the inho-mogeneities of media and non-uniformity of boundaries are taken into account in various real physical situations, the variable-coefficient nonlinear evolution equations (NLEEs) can often provide more powerful and realistic models than their constant-coefficient coun-terparts in describing a large variety of real phenomena. On the other hand, the Lie sym-metry analysis are more complicated than the constant-coefficient equations. The other highlight of the chapter is the generalized power series method and the method of unde-termined coefficient are utilized for tackling the exact explicit solutions.
In Chapter 7, the Painleve analysis and Lie symmetry methods are performed for the generalized KPP equation, Newell-Whitehead equation and its special case. These PDEs play an important role in nonlinear science and engineering, and they are mathematical models of many wave problems and mechanics. It is of importance also in biomathcmat-ics, etc.
Firstly, the Painleve property and the symmetries are obtained. Meanwhile, the exact solutions generated from the symmetry transformations are considered. Especially, the exact analytic solutions are investigated by the power series method. In this chapter, we considered three nonlinear evolution equations, the Painleve analysis is the highlight for this chapter. By the Painleve analysis method, the Backlund transformations and truncated expansion are obtained. Moreover, by the truncated expansion, the exact solutions are investigated for the three equations.
In Chapter 8, we summarize the main contents for the thesis firstly. Then, the points of innovation of this thesis are presented. At last, the prospect of further study for this direction are given.
本文以李(S.Lie)对称分析为基础和工具,综合运用动力系统的分支理论与方法、潘勒维尔(Painleve)分析、幂级数法(含推广的幂级数法)、待定系数法以及一些特殊的技巧与方法,研究偏微分方程的精确解及其相关的方程与解的性质。具体而言,即首先运用李对称分析得到方程的向量场或对称,然后利用相似约化将所研究的(非线性)偏微分方程化为常微分方程。这一步对方程而言可以说实现了实质性的转化,即把一个复杂的偏微分方程,包括各种非线性的、变系数的偏微分方程转化为一个常微分方程。接下来的工作就是研究这个常微分方程的解,求出了常微分方程的解,也就相应地得到了偏微分方程的解。这就是利用对称分析研究偏微分方程精确解的基本思路。当然,对称分析的作用远不止此,它与系统的可积性的研究还有着密切的关系,对称是系统本质属性的一种描述和刻画,它在偏微分方程与可积系统的研究中有着重要的意义与作用。这些我们将在研究偏微分方程精确解的同时一并加以介绍。至于如何研究约化得到的常微分方程,则主要涉及常微分方程与动力系统的理论与方法、幂级数法以及一些特殊的技巧与方法。
本文的主要内容如下：
第一章绪论。本章介绍了非线性科学的主要内容以及发展现状,综述了偏微分方程,尤其是非线性波方程的发展历史、研究现状、主要研究方法以及取得的主要成果。其中重点介绍了偏微分方程研究的主要方法,特别是对称分析在研究偏微分方程中的意义与作用。概括而言,这些方法各有特点,也都有各自的适用范围,都在特定的时期、特定的条件和各自的范围内发挥了应有的作用。有的方法可以说长盛不衰,历久弥新,至今还有强大的生命力,在偏微分方程的研究中仍然发挥着重要的作用。当然,任何一种方法都不是万能的,不会也不可能指望用一种方法解决所有的问题。本章的出发点是对各种主要的方法加以总结回顾,目的不是评判哪种方法的优劣,而是通过比较和总结,更好地继承和发扬其中蕴含的优秀的思想方法,从过去经典的思想与方法中汲取营养,更好地面向未来,进一步更深入地开展对现代偏微分方程及相关非线性科学的研究。
第二章理论准备。在这一章,列举了本文所涉及的一些相关知识,如李群与李代数、对称与向量场、向量场的延拓、Painleve分析简介、动力系统的分支理论与方法以及雅可比(Jacobi)椭圆函数等。限于篇幅,有些内容只列出主要概念与结论,详细内容可查阅后面的相关参考文献,此处不展开叙述。单列本章的目的是考虑到李群与对称分析的相关理论与知识比较多,通过本章,对有关的理论知识有所了解,便于后面的具体运用。
第三章基于李对称分析,研究了一般的Burgers'方程。该方程是一个既有非线性项又有二阶偏导项的非线性波方程,在理论和实践中有广泛的应用价值。它在一定条件下存在不同类型的孤波解,如冲击(震荡)波、稀疏波等。在流体力学、空气动力学的许多波动问题的研究中都要用到这个方程。例如在流体力学模型方程中,有线性Burgers'方程ut+aux=μuxx和非线性Burgers'方程ut+[f(u)]x=μuxx。当f(u)=1/2u2时,后者即为ut+uux=μuxx。在一定的初、边值条件下,可以得到这两类Burgers'方程的精确解,从而了解系统相应的流体力学性质。另外,Burgers'方程和许多重要的数学物理方程有着密切的联系,在非线性科学、流体力学以及工程技术中起着重要的基础性作用。在对称分析的基础上,首先求出了方程的群不变解以及任意次的迭代解。然后,利用对称约化将原方程化为各种形式的常微分方程,进而求出方程的精确解。其中应用了幂级数法(Power series method),得到了非线性、非自治的常微分方程严格的幂级数解,从而也就得到了相应的Burgers'方程的精确解,其中包含了不少新的显式精确解。
第四章研究推广的mKdV方程,众所周知,KdV方程是非常著名的浅水波方程,它起源于对水波问题的研究,KdV型方程可以描述各种浅水波的运动,在流体力学中有着广泛的应用。特别地,对于修正的KdV型方程,最近的研究发现可用于描述宇宙环境中超新星周围以及土星环的尘埃离子的波动规律,对于天体力学和大气物理的研究有着重要的意义。首先,通过对称分析得到了它的向量场。然后,由一般到特殊地得到了一些特殊而经典的KdV、mKdV方程的向量场。接下来,通过对称约化将推广的mKdV方程化为常微分方程,为下一步求解作准备。本章的一个亮点是运用了动力系统的分支理论与方法,详细全面地得到了推广的mKdV方程的显式精确解,包括幂级数解,同时还研究了系统的动力学性质。
第五章研究了一类短脉冲方程的精确解。短脉冲方程也是一类非常重要的非线性波方程,可以描述一些比较特殊的波。深入研究这类方程及其各种孤波解,对于了解一些特殊的波动问题具有重要意义。同时,该方程是一个重要的非线性数学物理方程,它在工程技术以及物理学、力学的许多领域都有重要应用。此方程不同于一般的非线性演化型方程,而是一个混合型的偏微分方程,这给对称分析带来了一定的困难。本章分别运用延拓法与待定系数法,得到了该方程的所有对称。其次,本章的另一特色是在运用动力系统的分支理论与方法研究方程的精确解时,引入了参数表示法,从而圆满地解决了解的显式表示问题。本章获得的这类短脉冲方程的精确解,都是用通常的方法难以得到的。
第六章研究了一类变系数债券方程。变系数偏微分方程最初主要来源于数学物理问题及大量的工程技术问题,但是,随着社会的进步和现代科学技术的不断发展,在各种经济社会领域、生物化学与环保领域、通讯信息与金融证券等领域,由于实际的需要也提出了越来越多的偏微分方程,这些方程一般形式复杂,且常常是变系数的。本章研究的变系数方程在金融数学与金融工程中经常用到,尤其是在期权定价问题的研究中,这类偏微分方程发挥着日益重要的作用。偏微分方程理论与现代经济、金融研究相结合,正成为一种重要的发展趋势。首先,对两个具体的变系数债券方程进行了对称分析,分别得出了它们的向量场。然后,又分别求出了它们的单参数群与群不变解。第三,利用相似变换分别将它们约化为常微分方程。第四,进一步求出它们的精确解。本章在内容上与前几章的主要不同之处在于,一是对称分析,由于所研究的方程是变系数的,因此,对称分析要比常系数方程复杂得多。二是在求精确解时除了幂级数法之外,还用了待定系数法等一些特殊方法,从而得到了方程的显式精确解,收到了较好的效果。三是在本章最后,我们还就一般形式的变系数债券方程进行了讨论,得出了它的对称及相应的精确解。
第七章研究了三个非线性演化方程。这类方程在非线性科学与工程技术中有着重要的意义与作用,是许多波动问题和力学问题的重要理论模型,在生物数学等领域也有着重要的应用。首先运用Painleve分析得到了它们的Painleve性质,以及相应的Backlund变换、截断展开式等。然后再通过对称分析,分别得到了它们的对称,并通过比较分析了Painleve分析与对称分析的异同。接着研究它们的精确解,除了基于对称分析的精确解,我们还得到了方程的基于Painleve截断展开的精确解。这些解的获得,是单独用任何一种方法所不可能得到的,这也说明了二者结合的意义和作用。另外,通过本章的研究可以发现,对于有些即使是不可积的方程,我们仍然可以利用对称分析与Painleve分析研究它们的精确解。我们知道,在可积系统的研究中,Painleve分析的主要作用是判断系统的可积性,但通过本章可以发现它还可以用于方程求解的研究。对称分析更是如此,无论是否可积,都可以通过对称分析研究方程的精确解。
总之,本文研究的对象是偏微分方程,包括各种非线性的、变系数的方程。主要目的是求出方程的解,尤其是显式的精确解。所以,本文所采用的方法与工具与一般孤子与可积系统的研究有所不同,结果也不一样,可以说各有侧重。限于论文的主题,尽管系统的对称与可积性如守恒律(CL)、Backlund变换等有着密切的联系,但对系统的可积性不作过多的讨论,目的是使论文主题更突出。另外,这些方程都是重要的数学物理方程,深入研究这些方程的解及其相关性质,如Painleve性质、可积性以及各种形式的解,尤其是各种显式精确解,对于了解系统所描述的具体问题的性质与规律,有着重要的意义与作用。
最后,在总结与展望中,首先概述了本文所获得的主要研究成果；然后,总结归纳了本文的主要创新点；最后,提出了围绕偏微分方程精确解的研究有待于进一步研究与思考的方向和问题。
Partial differential equations (PDEs), also called mathematical physics equations, are derived from the models of Physics, Mechanics and Engineering, etc. The earlier PDEs, such as Laplace equation and Poisson equation are based on the Newton's theory, wave equation, heat equation, and so on. And these equations are all classical PDEs. From the 19th century, a lot of new PDEs are derived, including the famous Maxwell system, Schrodinger equation, Einstein equation, Yang-Mills equation, and reaction-diffusion equation, etc. More and more PDEs, especially the nonlinear equations and systems will be derived owing to the progress of the modern science and technology. In addition, the nonlinear wave equations are important mathematical models for describing natural phe-nomena and one of the forefront topics in the studies of nonlinear mathematical physics, especially in the studies of soliton theory. The research on finding explicit and exact solu-tions of nonlinear wave equations and on analyzing the qualitative behavior of solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions, explain the corresponding natural phenomena reasonably, describe the essential properties of the nonlinear systems more deeply, and promote greatly the development of engineering technology and related subjects such as physics, mechanics and applied mathematics.
In this paper, the exact and explicit solutions, the dynamical behavior and the inte-grable properties for PDEs are investigated from the viewpoint of bifurcation theory of dynamical systems, Painleve analysis, (generalized) power series method, method of un-determined coefficient and a lot of special techniques, based on the Lie symmetry analysis. In detail, firstly, we solve the vector field or symmetries for a PDE with the Lie symmetry analysis method. Then, we reduce the PDE to the ordinary differential equations (ODEs) by the similarity transformations, it is a significant progress for dealing with the exact solutions of a PDE. That is, we transform a complicated PDE, including the nonlinear equation, variable coefficient equation, etc., to a ODE. Thirdly, we deal with the ODEs with the various method, such as the bifurcation theory of dynamical systems, Painleve analysis, (generalized) power series method, method of undetermined coefficient and a lot of special techniques. This is the outline for studying the exact solutions for PDEs. As we all know, by employing the Lie symmetry analysis, we can also consider the the integrable property for PDEs. But this is not the theme for this thesis, so we will not investigate it in detail.
The major works of this thesis are as follows.
Chapter 1 is the introduction. In this chapter, the historical background, research developments, main methods and achievements of partial differential equations especially the nonlinear wave equations are summarized. The discovery, corresponding research approaches and recent advance of Lie symmetry analysis and Painleve analysis methods are introduced. The relationship between partial differential equations and dynamical behavior, Painleve properties along with the study on exact solutions of partial differential equations by using the Lie symmetry analysis method are presented.
Chapter 2 is the preparation of theory. In this chapter, some preliminary knowledge of dynamical systems, theory of Lie groups and Lie algebras, prolongation of vector fields and the other basic mathematical theory and main results are introduced. This chapter is the preliminary knowledge of the thesis. For the sake of succinctness, we outline the summaries only. If it is necessary to learn some contents in detail, please consult the references.
In Chapter 3, the Lie symmetry analysis method is performed for the general Burg-ers'equation. This equation is a nonlinear wave equation, and it is of great importance in both theory and applications. The equation has various solitary wave solutions, such as the shock wave solutions, sparse wave solutions, etc., so it is important to study the wave problems in fluid dynamics and aerodynamics. For example, in the model equations of fluid dynamics, we have the linear Burgers' equation ut+aux=μuxx and the nonlinear Burgers'equation ut+[f(u)]x=μuxx, respectively. In particular, if f(u)=1/2u2,then the latter is ut+uux=μuxx.Under the certain conditions, we can get the exact solutions for the two equations. Thus, the fluid dynamical property can be comprehended. More-over, the Burgers'equation can be transformed into other important mathematical physics equations such as the heat equation, so it plays a significant role in the nonlinear science, fluid dynamics and engineering.
Based on the Lie symmetry analysis, we get the group-invariant solutions, iterated solutions. Then, by using the similarity reductions, we transform the general Burgers' equation to ordinary differential equations (ODEs), some of them are nonlinear and non- autonomous quations. Next, we deal with these ODEs by using the power series method and some special techniques, the exact solutions for the ODEs are obtained, so the solu-tions of the general Burgers'equation are presented simultaneously. Some new solutions are obtained for the first time in this chapter.
In Chapter 4, we investigate the extended mKdV equation. As is well known, the KdV equation is a famous shallow water wave equation, and it comes of the studies on the water wave problems. The KdV type equations can depict the various shallow water wave moment, and they play an important role in fluid dynamics. The modified KdV (mKdV) equation, on the other hand, has recently been discovered, e.g., to model the dust-ion-acoustic waves in such cosmic environments as those in the supernova shells and Saturn's F-ring, etc.
By using Lie symmetry analysis method, the vector field for the extended mKdV equation is obtained, and the vector fields of the several special classical KdV-type equa-tions are presented simultaneously. Then, by using the method of dynamical systems for the extended mKdV equation, all the exact solutions based on the Lie group method will be given. Especially, the bifurcations and traveling wave solutions are obtained. To guar-antee the existence of the above solutions, all parameter conditions are determined. Fur-thermore, the exact analytic solutions are considered by using the power series method. Such solutions for the equation are important in both applications and the theory of nonlin-ear science. Note that the vector fields of the several special classical KdV-type equations obtained in this way are a part of the vector fields rather than its complete vector fields, respectively. This reflect the intricacy of the Lie symmetry analysis. One of the highlight of the chapter is the method of dynamical systems. By employing this method, we discuss the dynamical behavior and get the all traveling wave solutions for the equation.
In Chapter 5, the Lie symmetry analysis and the generalized symmetry method are performed for a short pulse equation (SPE). The SPEs are nonlinear wave equations also, it can depict some peculiar waves. It is of great importance in comprehending some special wave problems that the SPEs and its solitary wave solutions are investigated in detail. At the same time, the SPEs are important mathematical physics equations, it play a significant role in engineering and mechanics.
Note that this equation is not a common evolution equation, but a mixed one. This brings some difficulty for the Lie symmetry analysis method. The symmetries for this equation are given. For the traveling wave solutions, the exact parametric representations are investigated. To guarantee the existence of the above solutions, all parameter condi-tions are determined. Furthermore, the exact analytic solutions are obtained by using the power series method. One of the highlight of the chapter is that we use the method of undetermined coefficient to solve the symmetries for the equation. Then, for getting the exact explicit solutions, the method of parameters are employed. It is not obtained by the general method for these solutions obtained in this chapter.
In Chapter 6, we use Lie symmetry group methods to study a pair of bond pricing equations. The variable-coefficient PDEs are derived from the mathematical physics and engineering problems at first. Now, many PDEs are derived from society, biochemistry, environmental protection, information science, communication, finance and securities, etc., along with the progress of society and the development of science and technology. Moreover, these PDEs are complicated and variable-coefficient usually. The variable-coefficient PDEs in this chapter are of importance in financial mathematics and financial engineering, and it play an important role in bond price as well.
The symmetries and similarity transformations for the two equations are provided, and all exact solutions are obtained. The general caseν≥2 are considered simultane-ously. We show that when the reduced equations are obtained, the generalized power se-ries method and special techniques can be used to find the exact solutions.When the inho-mogeneities of media and non-uniformity of boundaries are taken into account in various real physical situations, the variable-coefficient nonlinear evolution equations (NLEEs) can often provide more powerful and realistic models than their constant-coefficient coun-terparts in describing a large variety of real phenomena. On the other hand, the Lie sym-metry analysis are more complicated than the constant-coefficient equations. The other highlight of the chapter is the generalized power series method and the method of unde-termined coefficient are utilized for tackling the exact explicit solutions.
In Chapter 7, the Painleve analysis and Lie symmetry methods are performed for the generalized KPP equation, Newell-Whitehead equation and its special case. These PDEs play an important role in nonlinear science and engineering, and they are mathematical models of many wave problems and mechanics. It is of importance also in biomathcmat-ics, etc.
Firstly, the Painleve property and the symmetries are obtained. Meanwhile, the exact solutions generated from the symmetry transformations are considered. Especially, the exact analytic solutions are investigated by the power series method. In this chapter, we considered three nonlinear evolution equations, the Painleve analysis is the highlight for this chapter. By the Painleve analysis method, the Backlund transformations and truncated expansion are obtained. Moreover, by the truncated expansion, the exact solutions are investigated for the three equations.
In Chapter 8, we summarize the main contents for the thesis firstly. Then, the points of innovation of this thesis are presented. At last, the prospect of further study for this direction are given.
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