非局域对称及保对称离散格式的研究
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摘要
对称理论在数学物理等众多领域的研究中起着越来越重要的作用,已成为研究非线性系统可积性质及精确求解的有力工具之一.基于对称理论及符号计算,本文主要开展了以下三个方面的工作：构造了多个非线性数学物理方程的非局域对称,得到了这些方程的一些新的精确解；研究了非线性演化方程保对称离散方法,并将此方法推广到构造高维方程的保对称离散格式；基于符号计算平台Maple,开发了构造非线性微分方程非局域对称的自动推演程序包.本文的主要内容如下：
第一章作为绪论部分,重点介绍了对称理论、保对称离散、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.
第二章研究了若干非线性微分方程的非局域对称.首先,提出了一个辅助系统构造微分方程非局域对称的方法.将此方法运用到几类经典方程上,如Boussinesq方程、MKdV方程、AKNS系统等,得到了这些方程的非局域对称,并将非局域对称局域化到相应的封闭系统；其次,利用Lie群方法研究PIB方程,得到了约化方程的非局域对称,并利用对称约化方法构造了此方程新的精确解；最后,将基于对称构造非局域对称的方法应用到非线性diffusion-convection方程,得到了此方程非局域相关的势系统,并通过势系统构造了方程多个新的非局域对称.
第三章构造了几类经典方程的精确解.通过经典Lie群方法得到了封闭系统的对称群,构造了这些对称群的最优系统,并利用对称约化方法求得了这些方程一些新的精确解,如椭圆周期波和孤立子、椭圆周期波和扭结孤子相互作用解等.为了研究这些解的性质,对上述解选取适当参数并画出了相应的图像.
第四章研究了非线性发展方程的保对称离散格式.利用保对称离散方法构造了MKdV方程、Boussinesq方程的保对称离散格式；提出了一种构造高维非线性发展方程保对称离散格式的方法,并利用此方法构造了(2+1)维Burgers方程的保对称离散格式.通过验证可知,得到的离散格式均继承了连续方程的Lie点对称.
第五章开发了构造非线性微分方程非局域对称的程序包.基于微分方程的一些辅助系统,将提出的构造方程非局域对称的方法程序化,在Maple上编写相应的程序包NonSyml,并用多种不同类型的实例来说明此程序包的有效性,从而大大提高了构造非局域对称的效率.
第六章对全文工作进行讨论和总结,并对下一步要进行的研究工作做了展望.
Symmetry theory plays a more and more important role in the field of mathematical physics and so on, has become one of powerful tools in studying integrable properties and exact solutions of nonlinear system. Based on symmetry theory and symbolic com-putation, the main work is carried out in three aspects:nonlocal symmetries of nonlinear differential equations are investigated in this dissertation, some new exact solutions of these equations are obtained; Symmetry-preserving discrete, algorithm of nonlinear evo-lution equations is studied, and the method is extended to higher dimensional equation-s; Furthermore, based on symbolic computation platform Maple, a software package of nonlocal symmetry automatic deduction is developed. The main work is carried out as follows:
In chapter1, an introduction of the research background and the current situation re-view related to this dissertation, which including symmetry theory, symmetry-preserving discrete and symbolic computation is devoted. The main works of this dissertation are also illustrated.
In chapter2, nonlocal symmetries of several differential equations are studied. First-1y, based on the auxiliary systems, a method of constructing nonlocal symmetry is pro-posed, and the method is applied to several classical equations, such as Boussinesq equa-tion, MKdV equation, AKNS system, etc. The nonlocal symmetries of these equations are obtained, and localized to corresponding closed systems. Secondly, we study PIB equation by using Lie group method and construct nonlocal symmetries of reduced e-quations. Some new exact solutions.of the PIB equation are obtained by using symmetry reduction method. Finally, nonlinear diffusion-convection equation is studied by using symmetry-based method, nonlocal-related potential systems and nonlocal symmetries are constructed.
In chapter3, exact solutions of several classical equation are constructed. The sym-metry groups of the closed systems are obtained through classical Lie group method, op-timal systems of the symmetry groups are constructed. Some new exact solutions of these equations are obtained by using the symmetry reduction method, among which the inter-actions of elliptic periodic wave and soliton as well as periodic wave and kink soliton are found. In order to study the properties of the solutions, we select appropriate parameters and draw the corresponding images of above solutions.
In chapter4, symmetry-preserving discrete format of nonlinear evolution equations is discussed. Symmetry-preserving discrete formats of MKdV equation, Boussinesq e- quation are constructed by using symmetry-preserving discrete method. We put forward a method which can construct symmetry-preserving discrete formats of high dimension-al nonlinear evolution equations, and apply this method to (2+1)-dimensional Burgers equation. Corresponding symmetry-preserving discrete formats which inherit Lie point symmetries of continuous equations are obtained.
In chapter5, a software package of constructing nonlocal symmetries of PDEs is developed. Based on some auxiliary systems of PDEs, we put forward a mechaniza-tion method of constructing nonlocal symmetries and develop the corresponding software package NonSymI. Multiple instances are used to prove the effectiveness of this package, thus the efficiency of constructing nonlocal symmetries is improved greatly.
In chapter6, the summary and discussion of this dissertation are given, and the outlook of future works are discussed.
第一章作为绪论部分,重点介绍了对称理论、保对称离散、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.
第二章研究了若干非线性微分方程的非局域对称.首先,提出了一个辅助系统构造微分方程非局域对称的方法.将此方法运用到几类经典方程上,如Boussinesq方程、MKdV方程、AKNS系统等,得到了这些方程的非局域对称,并将非局域对称局域化到相应的封闭系统；其次,利用Lie群方法研究PIB方程,得到了约化方程的非局域对称,并利用对称约化方法构造了此方程新的精确解；最后,将基于对称构造非局域对称的方法应用到非线性diffusion-convection方程,得到了此方程非局域相关的势系统,并通过势系统构造了方程多个新的非局域对称.
第三章构造了几类经典方程的精确解.通过经典Lie群方法得到了封闭系统的对称群,构造了这些对称群的最优系统,并利用对称约化方法求得了这些方程一些新的精确解,如椭圆周期波和孤立子、椭圆周期波和扭结孤子相互作用解等.为了研究这些解的性质,对上述解选取适当参数并画出了相应的图像.
第四章研究了非线性发展方程的保对称离散格式.利用保对称离散方法构造了MKdV方程、Boussinesq方程的保对称离散格式；提出了一种构造高维非线性发展方程保对称离散格式的方法,并利用此方法构造了(2+1)维Burgers方程的保对称离散格式.通过验证可知,得到的离散格式均继承了连续方程的Lie点对称.
第五章开发了构造非线性微分方程非局域对称的程序包.基于微分方程的一些辅助系统,将提出的构造方程非局域对称的方法程序化,在Maple上编写相应的程序包NonSyml,并用多种不同类型的实例来说明此程序包的有效性,从而大大提高了构造非局域对称的效率.
第六章对全文工作进行讨论和总结,并对下一步要进行的研究工作做了展望.
Symmetry theory plays a more and more important role in the field of mathematical physics and so on, has become one of powerful tools in studying integrable properties and exact solutions of nonlinear system. Based on symmetry theory and symbolic com-putation, the main work is carried out in three aspects:nonlocal symmetries of nonlinear differential equations are investigated in this dissertation, some new exact solutions of these equations are obtained; Symmetry-preserving discrete, algorithm of nonlinear evo-lution equations is studied, and the method is extended to higher dimensional equation-s; Furthermore, based on symbolic computation platform Maple, a software package of nonlocal symmetry automatic deduction is developed. The main work is carried out as follows:
In chapter1, an introduction of the research background and the current situation re-view related to this dissertation, which including symmetry theory, symmetry-preserving discrete and symbolic computation is devoted. The main works of this dissertation are also illustrated.
In chapter2, nonlocal symmetries of several differential equations are studied. First-1y, based on the auxiliary systems, a method of constructing nonlocal symmetry is pro-posed, and the method is applied to several classical equations, such as Boussinesq equa-tion, MKdV equation, AKNS system, etc. The nonlocal symmetries of these equations are obtained, and localized to corresponding closed systems. Secondly, we study PIB equation by using Lie group method and construct nonlocal symmetries of reduced e-quations. Some new exact solutions.of the PIB equation are obtained by using symmetry reduction method. Finally, nonlinear diffusion-convection equation is studied by using symmetry-based method, nonlocal-related potential systems and nonlocal symmetries are constructed.
In chapter3, exact solutions of several classical equation are constructed. The sym-metry groups of the closed systems are obtained through classical Lie group method, op-timal systems of the symmetry groups are constructed. Some new exact solutions of these equations are obtained by using the symmetry reduction method, among which the inter-actions of elliptic periodic wave and soliton as well as periodic wave and kink soliton are found. In order to study the properties of the solutions, we select appropriate parameters and draw the corresponding images of above solutions.
In chapter4, symmetry-preserving discrete format of nonlinear evolution equations is discussed. Symmetry-preserving discrete formats of MKdV equation, Boussinesq e- quation are constructed by using symmetry-preserving discrete method. We put forward a method which can construct symmetry-preserving discrete formats of high dimension-al nonlinear evolution equations, and apply this method to (2+1)-dimensional Burgers equation. Corresponding symmetry-preserving discrete formats which inherit Lie point symmetries of continuous equations are obtained.
In chapter5, a software package of constructing nonlocal symmetries of PDEs is developed. Based on some auxiliary systems of PDEs, we put forward a mechaniza-tion method of constructing nonlocal symmetries and develop the corresponding software package NonSymI. Multiple instances are used to prove the effectiveness of this package, thus the efficiency of constructing nonlocal symmetries is improved greatly.
In chapter6, the summary and discussion of this dissertation are given, and the outlook of future works are discussed.
引文
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