非线性波、几何可积性与群分类
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摘要
本文以著名数学家吴文俊先生所倡导的数学机械化思想为指导，以构造性的变换及符号计算为辅助工具，从几何和代数的角度来研究了非线性波，可积系统和微分方程的群理论分析中的一些问题：精确波解(行波解、孤立波解、周期解、泛函分离变量解)、Darboux变换、非等谱演化方程与几何可积性、群分类、等价群，等价性变换、古典李对称约化、守恒律分类。
第二和第三章主要考虑了非线性偏微分方程的精确解的构造。首先介绍了张鸿庆教授提出的构造非线性偏微分方程精确解的AC=BD模式和C-D对理论，并且把这一模式推广到研究(1+1)-维偏微分方程的保持形式的点变换。然后在第三章具体研究了这一模式的应用：(ⅰ)基于一类一阶带六次非线性项的常微分方程，提出了扩展的第一类椭圆方程方法，并以广义的Zakharov方程组为例来展示该方法的有效性，获得了大量新的有趣的精确解，其中包括钟型和扭结型孤波解，亮和暗孤立波解，三角周期波解等；(ⅱ)基于一类投影Riccati方程，提出了一种新的变系数投影Riccati方程展开法。利用该方法，获得(2+1)-维广义Broer-Kaup方程的许多有趣的新的类孤波解和有理解，当把解中的某些任意函数取为行波变换时，还可得到许多具有重要物理意义的行波解；(ⅲ)构造了四类(1+1)-维孤子方程的三种显式的N-重Darboux变换，利用这些变换，获得了它们以及(2+1)-维Kadomtsev-Petviashvili方程和修正的Kadomtsev-Petviashvili方程的有趣的(2N-1)和(2N)-孤子解，而且所有的变换和解都用类-Vandermonde行列式表示，使得其形式相当的简洁。
近年来，人们对孤子和可积系统理论中的非等谱演化方程，即其相应谱问题具有时间依赖的谱参数η，越来越感兴趣。第四章给出了两类演化方程u_t=F(x，t，u，u_x，…，u_x~k)和u_(xt)=G(x，t，u，u_x，…，u_x~k)在假设η为x，t的可微函数下描述伪球曲面(几何可积)的完整刻画。因此提供了一个系统的程序确定一个非等谱线性问题，使得它是给定的非等谱演化方程的可积性条件。从而为解决可积系统理论中的核心问题之一：给定一个非线性微分方程，判断它是否Lax意义下可积，即是否可写成一对线性问题的可积性条件提供了一种重要的几何途径。上述内容形成本文的第一部分。
微分方程的群分类，特别是完备的群分类是微分方程群理论分析领域经典而又非常困难的问题之一。第五章利用相容性方法以及附加的等价性变换，给出了一类带有变系数函数f的(1+1)-维非线性电报方程f(x)u_(tt)=(H(u)u_x)_x+K(u)u_x的完备群分类。结果获得了大量新的有趣的具有非平凡变系数函数的非线性不变模型，它们都具有非平凡的对称代数，而且这些对称代数至多是五维的。作为上述分类结果的应用，还给出了非线性电报方程u_(tt)=(H(u)u_x)_x+K(u)u_x的完备群分类。另外，还研究了所有不变模型的附加的等价性变换，并且通过利用这些附加的等价性变换，古典Lie约化方法以及一般条件对称方法，给出了某些特殊的变系数非线性不变模型与非线性电报方程的精确解和泛函分离变量解。最后还给出该类变系数非线性电报方程在等价性变换群下具有零阶特征的局部守恒律的分类。
第六章利用古典无穷小算法，等价性变换技巧和低维抽象李代数的分类理论给出了一般KdV-类非线性演化方程u_t=F(t，x，u，u_x，u_(xx))u_(xxx)+G(t，x，u，u_x，u_(xx))在四维及四维以下李代数下不变的群分类。证明了只存在三个不等价的方程在三维单李代数下不变，而且进一步证明在所有半单李代数下不变的不等价方程只有这三个。另外，还证明了存在两个，五个，二十九个和二十六个不等价的方程分别在一维，二维，三维和四维可解李代数下不变。第五和第六章形成了本文的第二部分。
In this dissertation, under the guidance of mathematical mechanization proposed by famousmathematician Wu Wentsun and by means of many types of constructive transformations aswell as symbolic computation, some topics in nonlinear waves, integrable systems and grouptheoretical analysis of differential equations are studied from the points of view of geometry andalgebra, including exact solutions, Darboux transformations, non-isospectral evolution equationswhich describe pseudo-sphere surfaces, symmetries group classifications, additional equivalencetransformations, classical Lie reduction and classifications of conservation laws.
Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear partial differentialequations. Firstly, the basic theories of AC=BD model and C-D pair are introduced, and thenthey are extended to studying the form-preserving transformations of (1+1)-dimensional partialdifferential equations. Secondly, we choose some examples to illustrate them in Chapter 3. (ⅰ)Based on a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term,an extended first kind elliptic sub-equation method is proposed to obtain solutions of nonlineardifferential equations. Many interesting exact solutions of generalized Zakharov equations are ex-plored, including new bell and kink profile solitary wave solutions, bright and dark solitary wavesolutions, triangular periodic wave solutions and singular solutions; (ⅱ) a variable-coefficientprojective Riccati equation method is presented to obtain non-travelling wave solutions for the(2+1)-dimensional generalized Broer-Kaup system; (ⅲ) Three kinds of explicit N-fold Dar-boux transformation of four (1+1)-dimensional soliton systems are constructed. Then thesetransformations are used to derive explicit (2N-1) and (2N)-soliton solutions of these systemsand the (2+1)-dimensional Kadomtsev-Petviashvili equation as well as modified Kadomtsev-Petviashvili equation. The explicit formulas of both the Darboux transformations and solitonsolutions are expressed by Vandermonde-like determinants which are remarkable compactnessand transparency.
Recent years there are increasing interests in non-isospectral evolution equation, i. e.,the corresponding spectral problem with a time-dependent spectral parameterη, in the the-ory of soliton and integrable system. In chapter 4, characterizations of evolution equationsu_t=F(x, t, u, u_x, ..., u_x~k) and u_(xt)=F(x, t, u, u_x, ..., u_x~k) which describe pseudo-spherical surfacesare given, under a priori assumption thatηis differential function of x, t, thus providing asystematic procedure to determine a non-isospectral linear problem for which the given non- isospectral evolution equation is the integrability condition. It also paves a way form the pointof view of geometry to solve one of the central problems of integrable systems: To determinea given nonlinear differential equation integrable or not in Lax sense, i. e., whether it can bewritten as an integrablity condition of a pair of linear problems. The aforementioned subjectsform the first part of this dissertation.
Group classification of differential equations, especially complete group classification, isone of the classical and very tough problems in the field of group theoretical analysis of dif-ferential equation. In chapter 5, complete group classification of a class of variable coefficient(1+1)-dimensional nonlinear telegraph equations f(x)u_(tt)=(H(u)u_x)_x+K(u)u_x, is given, byusing a compatibility method and additional equivalence transformations. A number of newinteresting nonlinear invariant models which have non-trivial symmetry algebra are obtained. Itis shown that the symmetry algebra is at most five-dimensional. As an application, the groupclassification of nonlinear telegraph equations u_(tt)=(H(u)u_x)_x+K(u)u_x is also provided. Fur-thermore, the possible additional equivalence transformations between equations from the classunder consideration are investigated. Exact solutions of special forms of these equations arealso constructed via classical Lie method and generalized conditional transformations. Localconservation laws with characteristics of order 0 of the class under consideration are classifiedwith respect to the group of equivalence transformations.
Chapter 6 deals with the group classification of general KdV-type nonlinear evolutionequations of the form u_t=F(t, x, u, u_x, u_(xx))u_(xxx)+a(t, x, u, u_x, u_(xx)) invariant under at mostfour-dimensional Lie algebra, by using the classical infinitesimal Lie method, the technique ofequivalence transformations and the theory of classification of abstract low-dimensional Lie alge-bras. It is shown that there are three equations admitting three dimensional simple Lie algebras,what's more, all the inequivalent equations admitting simple Lie algebra are nothing but them.Furthermore, we prove that there exist two, five, twenty-nine and twenty-six inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvableLie algebras, respectively. Chapter 5 and 6 form the second part of this dissertation.
第二和第三章主要考虑了非线性偏微分方程的精确解的构造。首先介绍了张鸿庆教授提出的构造非线性偏微分方程精确解的AC=BD模式和C-D对理论，并且把这一模式推广到研究(1+1)-维偏微分方程的保持形式的点变换。然后在第三章具体研究了这一模式的应用：(ⅰ)基于一类一阶带六次非线性项的常微分方程，提出了扩展的第一类椭圆方程方法，并以广义的Zakharov方程组为例来展示该方法的有效性，获得了大量新的有趣的精确解，其中包括钟型和扭结型孤波解，亮和暗孤立波解，三角周期波解等；(ⅱ)基于一类投影Riccati方程，提出了一种新的变系数投影Riccati方程展开法。利用该方法，获得(2+1)-维广义Broer-Kaup方程的许多有趣的新的类孤波解和有理解，当把解中的某些任意函数取为行波变换时，还可得到许多具有重要物理意义的行波解；(ⅲ)构造了四类(1+1)-维孤子方程的三种显式的N-重Darboux变换，利用这些变换，获得了它们以及(2+1)-维Kadomtsev-Petviashvili方程和修正的Kadomtsev-Petviashvili方程的有趣的(2N-1)和(2N)-孤子解，而且所有的变换和解都用类-Vandermonde行列式表示，使得其形式相当的简洁。
近年来，人们对孤子和可积系统理论中的非等谱演化方程，即其相应谱问题具有时间依赖的谱参数η，越来越感兴趣。第四章给出了两类演化方程u_t=F(x，t，u，u_x，…，u_x~k)和u_(xt)=G(x，t，u，u_x，…，u_x~k)在假设η为x，t的可微函数下描述伪球曲面(几何可积)的完整刻画。因此提供了一个系统的程序确定一个非等谱线性问题，使得它是给定的非等谱演化方程的可积性条件。从而为解决可积系统理论中的核心问题之一：给定一个非线性微分方程，判断它是否Lax意义下可积，即是否可写成一对线性问题的可积性条件提供了一种重要的几何途径。上述内容形成本文的第一部分。
微分方程的群分类，特别是完备的群分类是微分方程群理论分析领域经典而又非常困难的问题之一。第五章利用相容性方法以及附加的等价性变换，给出了一类带有变系数函数f的(1+1)-维非线性电报方程f(x)u_(tt)=(H(u)u_x)_x+K(u)u_x的完备群分类。结果获得了大量新的有趣的具有非平凡变系数函数的非线性不变模型，它们都具有非平凡的对称代数，而且这些对称代数至多是五维的。作为上述分类结果的应用，还给出了非线性电报方程u_(tt)=(H(u)u_x)_x+K(u)u_x的完备群分类。另外，还研究了所有不变模型的附加的等价性变换，并且通过利用这些附加的等价性变换，古典Lie约化方法以及一般条件对称方法，给出了某些特殊的变系数非线性不变模型与非线性电报方程的精确解和泛函分离变量解。最后还给出该类变系数非线性电报方程在等价性变换群下具有零阶特征的局部守恒律的分类。
第六章利用古典无穷小算法，等价性变换技巧和低维抽象李代数的分类理论给出了一般KdV-类非线性演化方程u_t=F(t，x，u，u_x，u_(xx))u_(xxx)+G(t，x，u，u_x，u_(xx))在四维及四维以下李代数下不变的群分类。证明了只存在三个不等价的方程在三维单李代数下不变，而且进一步证明在所有半单李代数下不变的不等价方程只有这三个。另外，还证明了存在两个，五个，二十九个和二十六个不等价的方程分别在一维，二维，三维和四维可解李代数下不变。第五和第六章形成了本文的第二部分。
In this dissertation, under the guidance of mathematical mechanization proposed by famousmathematician Wu Wentsun and by means of many types of constructive transformations aswell as symbolic computation, some topics in nonlinear waves, integrable systems and grouptheoretical analysis of differential equations are studied from the points of view of geometry andalgebra, including exact solutions, Darboux transformations, non-isospectral evolution equationswhich describe pseudo-sphere surfaces, symmetries group classifications, additional equivalencetransformations, classical Lie reduction and classifications of conservation laws.
Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear partial differentialequations. Firstly, the basic theories of AC=BD model and C-D pair are introduced, and thenthey are extended to studying the form-preserving transformations of (1+1)-dimensional partialdifferential equations. Secondly, we choose some examples to illustrate them in Chapter 3. (ⅰ)Based on a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term,an extended first kind elliptic sub-equation method is proposed to obtain solutions of nonlineardifferential equations. Many interesting exact solutions of generalized Zakharov equations are ex-plored, including new bell and kink profile solitary wave solutions, bright and dark solitary wavesolutions, triangular periodic wave solutions and singular solutions; (ⅱ) a variable-coefficientprojective Riccati equation method is presented to obtain non-travelling wave solutions for the(2+1)-dimensional generalized Broer-Kaup system; (ⅲ) Three kinds of explicit N-fold Dar-boux transformation of four (1+1)-dimensional soliton systems are constructed. Then thesetransformations are used to derive explicit (2N-1) and (2N)-soliton solutions of these systemsand the (2+1)-dimensional Kadomtsev-Petviashvili equation as well as modified Kadomtsev-Petviashvili equation. The explicit formulas of both the Darboux transformations and solitonsolutions are expressed by Vandermonde-like determinants which are remarkable compactnessand transparency.
Recent years there are increasing interests in non-isospectral evolution equation, i. e.,the corresponding spectral problem with a time-dependent spectral parameterη, in the the-ory of soliton and integrable system. In chapter 4, characterizations of evolution equationsu_t=F(x, t, u, u_x, ..., u_x~k) and u_(xt)=F(x, t, u, u_x, ..., u_x~k) which describe pseudo-spherical surfacesare given, under a priori assumption thatηis differential function of x, t, thus providing asystematic procedure to determine a non-isospectral linear problem for which the given non- isospectral evolution equation is the integrability condition. It also paves a way form the pointof view of geometry to solve one of the central problems of integrable systems: To determinea given nonlinear differential equation integrable or not in Lax sense, i. e., whether it can bewritten as an integrablity condition of a pair of linear problems. The aforementioned subjectsform the first part of this dissertation.
Group classification of differential equations, especially complete group classification, isone of the classical and very tough problems in the field of group theoretical analysis of dif-ferential equation. In chapter 5, complete group classification of a class of variable coefficient(1+1)-dimensional nonlinear telegraph equations f(x)u_(tt)=(H(u)u_x)_x+K(u)u_x, is given, byusing a compatibility method and additional equivalence transformations. A number of newinteresting nonlinear invariant models which have non-trivial symmetry algebra are obtained. Itis shown that the symmetry algebra is at most five-dimensional. As an application, the groupclassification of nonlinear telegraph equations u_(tt)=(H(u)u_x)_x+K(u)u_x is also provided. Fur-thermore, the possible additional equivalence transformations between equations from the classunder consideration are investigated. Exact solutions of special forms of these equations arealso constructed via classical Lie method and generalized conditional transformations. Localconservation laws with characteristics of order 0 of the class under consideration are classifiedwith respect to the group of equivalence transformations.
Chapter 6 deals with the group classification of general KdV-type nonlinear evolutionequations of the form u_t=F(t, x, u, u_x, u_(xx))u_(xxx)+a(t, x, u, u_x, u_(xx)) invariant under at mostfour-dimensional Lie algebra, by using the classical infinitesimal Lie method, the technique ofequivalence transformations and the theory of classification of abstract low-dimensional Lie alge-bras. It is shown that there are three equations admitting three dimensional simple Lie algebras,what's more, all the inequivalent equations admitting simple Lie algebra are nothing but them.Furthermore, we prove that there exist two, five, twenty-nine and twenty-six inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvableLie algebras, respectively. Chapter 5 and 6 form the second part of this dissertation.
引文
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