非线性系统的对称性与可积性
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摘要
本文基于符号计算,研究了非线性数学物理中的对称性和可积性理论及其应用问题。主要开展了四个方面的工作：应用经典对称方法研究了流体力学和大气海洋中的一些重要的非线性模型；发展了非局域对称的理论和相关应用；利用Bell多项式系统地分析了含任意参数的非线性发展方程的可积性质；开发了构造有限维Lie代数一维最优系统的程序包。
第一章作为绪论部分,重点介绍了对称理论、可积系统和符号计算的背景与发展现状,并且阐明了本论文的主要工作。
第二章研究了经典对称理论在流体力学和大气海洋动力学中的应用。首先利用一些经典对称方法讨论了流体力学中与非线性薛定谔方程相关的一个新型非线性模型：共振DS系统,分析了其连续和离散的对称性质,给出了三种类型的相似约化；其次,研究了大气海洋中的一个重要的非线性高维模型—(3+1)维的斜压位涡方程,从相关低维方程的解出发构造了此高维方程丰富的严格解,解释了大气运动中带有丰富垂直结构的Rossby波和一些经典的径向环流现象。
第三章发展了非局域的对称理论和相关应用。将发展的非局域对称理论应用于不同形式的KdV方程,分别通过对Backlund变换、Darboux变换和双线性变换取无穷小形式得到了不同形式的非局域对称。一方面,成功构造了KdV方程丰富的精确解,并首次发现了KdV方程椭圆周期波和孤立子的相互作用以及Painleve波和孤立子的相互作用的严格解表达式；另一方面,给出了势KdV方程的负可积梯队和其他新的可积系统。
第四章讨论了Bell多项式和相关可积性。将双Bell多项式方法推广到含四个任意参数的广义NNV方程和非等谱变系数的mKdV方程,系统地给出了方程的Hirota双线性形式、双线性Backlund变换、Lax对和无穷多守恒律。这种代数方法只用到维数分析和基本组合知识,不需要很多技巧性的猜测工作。
第五章考虑了对称理论中一维最优系统的算法问题。针对有限维Lie代数一维最优系统的构造,我们通过总结和分析看似随意和经验性的分类过程,首次给出了一个机械化算法,并在Maple上开发了相应的算法实现程序包,从而节省了大量人工重复繁琐的计算。最后以实例说明了该算法的有效性和实用性。
第六章对全文的工作进行了总结和概括,并对下一步需要进行的研究工作进行了展望。
Based on symbolic computation, this dissertation investigates the theory with appli-cation of the symmetry and integrability of nonlinear mathematical physics. The main work is carried out in four aspects:the classical symmetry methods are applied to study some important nonlinear models in fluid mechanics as well as the atmosphere and ocean; the theory of nonlocal symmetry is developed and some related applications are real-ized; Bell polynomials are used to analyze the integrability of nonlinear evolution equa-tions with arbitrary parameters; a package of one-dimensional optimal system for finite-dimensional Lie algebra is developed.
Chapter1is an introduction to review the theoretical background and development of symmetry theory, integrable system and symbolic computation. The main works of this dissertation are also illustrated.
Chapter2concentrates on applying the classical symmetry theory in fluid mechanics as well as atmospheric and oceanic dynamics. Firstly, a novel nonlinear model in fluid mechanics called resonant Davey-Stewartson (DS) system, which is relevant to nonlinear Schrodinger equation, is investigated by some classical symmetry methods. Its continuous and discrete symmetry properties are discussed and three types of reduced equations are obtained. Then, an important nonlinear high-dimensional model in the atmospheric and oceanic dynamics, namely (3+1)-dimensional baroclinic potential vorticity equation is studied. Kinds of exact solutions are obtained from its corresponding lower-dimensional equation. According to these explicit solutions, the Rossby wave with rich vertical struc-ture of the atmospheric motion and some classical radial circulation phenomenons are explained.
Chapter3is devoted to developing a new theory of nonlocal symmetry and related applications. Applying the developed nonlocal symmetry method to KdV equation in different forms, many nonlocal symmetries are obtained by taking infinitesimal forms from Backlund transformation, Darboux transformation and bilinear transformation re-spectively. On one hand, abundant explicit solutions of KdV equation are constructed successfully, among which the interactions of elliptic periodic wave and soliton as well as Painleve wave and soliton are firstly discovered. On the other hand, the negative hier-archies of potential KdV equation and other new integrable systems are presented.
Chapter4deals with Bell polynomials and related integrabilities. The binary Bell polynomials method is extended to a generalized NNV equation with four arbitrary pa-rameters as well as a nonisospectral and variable-coefficient mKdV equation respectively. Their corresponding Hirota bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws are obtained step by step. This algebraic approach only considers dimensional analysis and elementary combinatorics, without too much clever guesswork.
Chapter5focuses on the algorithm of one-dimensional optimal system in symmetry theory. The mechanical algorithm for constructing one-dimensional optimal system of finite-dimensional Lie algebra is firstly provided and corresponding implementation of software package on Maple is accomplished. We summarize and analyze the seemingly random and empirical classification process to design a rule which can be achieved on the computer. Due to our algorithm, a large number of repetitive and tedious calculations are avoided. Lastly, two examples are given to illustrate the effectiveness and practicality of this algorithm.
Chapter6concerns the summary and discussion of the whole dissertation, and the prospect for the future work is also put forward.
第一章作为绪论部分,重点介绍了对称理论、可积系统和符号计算的背景与发展现状,并且阐明了本论文的主要工作。
第二章研究了经典对称理论在流体力学和大气海洋动力学中的应用。首先利用一些经典对称方法讨论了流体力学中与非线性薛定谔方程相关的一个新型非线性模型：共振DS系统,分析了其连续和离散的对称性质,给出了三种类型的相似约化；其次,研究了大气海洋中的一个重要的非线性高维模型—(3+1)维的斜压位涡方程,从相关低维方程的解出发构造了此高维方程丰富的严格解,解释了大气运动中带有丰富垂直结构的Rossby波和一些经典的径向环流现象。
第三章发展了非局域的对称理论和相关应用。将发展的非局域对称理论应用于不同形式的KdV方程,分别通过对Backlund变换、Darboux变换和双线性变换取无穷小形式得到了不同形式的非局域对称。一方面,成功构造了KdV方程丰富的精确解,并首次发现了KdV方程椭圆周期波和孤立子的相互作用以及Painleve波和孤立子的相互作用的严格解表达式；另一方面,给出了势KdV方程的负可积梯队和其他新的可积系统。
第四章讨论了Bell多项式和相关可积性。将双Bell多项式方法推广到含四个任意参数的广义NNV方程和非等谱变系数的mKdV方程,系统地给出了方程的Hirota双线性形式、双线性Backlund变换、Lax对和无穷多守恒律。这种代数方法只用到维数分析和基本组合知识,不需要很多技巧性的猜测工作。
第五章考虑了对称理论中一维最优系统的算法问题。针对有限维Lie代数一维最优系统的构造,我们通过总结和分析看似随意和经验性的分类过程,首次给出了一个机械化算法,并在Maple上开发了相应的算法实现程序包,从而节省了大量人工重复繁琐的计算。最后以实例说明了该算法的有效性和实用性。
第六章对全文的工作进行了总结和概括,并对下一步需要进行的研究工作进行了展望。
Based on symbolic computation, this dissertation investigates the theory with appli-cation of the symmetry and integrability of nonlinear mathematical physics. The main work is carried out in four aspects:the classical symmetry methods are applied to study some important nonlinear models in fluid mechanics as well as the atmosphere and ocean; the theory of nonlocal symmetry is developed and some related applications are real-ized; Bell polynomials are used to analyze the integrability of nonlinear evolution equa-tions with arbitrary parameters; a package of one-dimensional optimal system for finite-dimensional Lie algebra is developed.
Chapter1is an introduction to review the theoretical background and development of symmetry theory, integrable system and symbolic computation. The main works of this dissertation are also illustrated.
Chapter2concentrates on applying the classical symmetry theory in fluid mechanics as well as atmospheric and oceanic dynamics. Firstly, a novel nonlinear model in fluid mechanics called resonant Davey-Stewartson (DS) system, which is relevant to nonlinear Schrodinger equation, is investigated by some classical symmetry methods. Its continuous and discrete symmetry properties are discussed and three types of reduced equations are obtained. Then, an important nonlinear high-dimensional model in the atmospheric and oceanic dynamics, namely (3+1)-dimensional baroclinic potential vorticity equation is studied. Kinds of exact solutions are obtained from its corresponding lower-dimensional equation. According to these explicit solutions, the Rossby wave with rich vertical struc-ture of the atmospheric motion and some classical radial circulation phenomenons are explained.
Chapter3is devoted to developing a new theory of nonlocal symmetry and related applications. Applying the developed nonlocal symmetry method to KdV equation in different forms, many nonlocal symmetries are obtained by taking infinitesimal forms from Backlund transformation, Darboux transformation and bilinear transformation re-spectively. On one hand, abundant explicit solutions of KdV equation are constructed successfully, among which the interactions of elliptic periodic wave and soliton as well as Painleve wave and soliton are firstly discovered. On the other hand, the negative hier-archies of potential KdV equation and other new integrable systems are presented.
Chapter4deals with Bell polynomials and related integrabilities. The binary Bell polynomials method is extended to a generalized NNV equation with four arbitrary pa-rameters as well as a nonisospectral and variable-coefficient mKdV equation respectively. Their corresponding Hirota bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws are obtained step by step. This algebraic approach only considers dimensional analysis and elementary combinatorics, without too much clever guesswork.
Chapter5focuses on the algorithm of one-dimensional optimal system in symmetry theory. The mechanical algorithm for constructing one-dimensional optimal system of finite-dimensional Lie algebra is firstly provided and corresponding implementation of software package on Maple is accomplished. We summarize and analyze the seemingly random and empirical classification process to design a rule which can be achieved on the computer. Due to our algorithm, a large number of repetitive and tedious calculations are avoided. Lastly, two examples are given to illustrate the effectiveness and practicality of this algorithm.
Chapter6concerns the summary and discussion of the whole dissertation, and the prospect for the future work is also put forward.
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