微分方程的精确解、群与群不变解的分类问题
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摘要
本文以数学机械化思想和AC=BD模式为指导,研究具有任意阶非线性项的非线性偏微分方程的精确求解,微分差分方程的群分类和超对称方程的群不变解的分类问题.
第一章介绍数学机械化、孤立子理论、数学物理方程的精确求解、对称分析、群分类、超对称和超对称方程的历史发展和研究现状,并介绍本文的选题及主要工作.
第二章介绍微分方程变换的机械化构造的AC=BD理论和C-D对理论的基本内容和思想.
第三章基于将非线性发展方程精确求解代数化、算法化、机械化的指导思想和AC=BD理论,改进广义Riccati方程有理展开法,并推广Sub-ODE方法,进而给出带有任意阶非线性项的非线性发展方程更多的精确解.
第四章利用Zhdanov和Lahno给出的求解偏微分方程群分类的方法研究非线性微分差分方程(?)_n=F_n(t,u_(n-1),u_n,u_(n+1))在李代数下不变的群分类.
第五章给出超对称二玻色子方程的李超代数的伴随表示关系及其在这种关系下一维子代数的共轭类,进一步计算群不变解的初步分类,并得到超对称二玻色子方程的指数函数解、三角函数解和有理解.
In this dissertation, some topics axe studied with the idea of mathematical mechanizationand AC=BD model, including exact solutions of nonlinear partial differential equations with nonlinear terms of any order, group classification of differential-difference equations and classification of the group-invariant solutions of supersymmetric equations.
In Chapter 1, we introduce the history and development of mathematical mechanization,soliton theory, solutions of the mathematical and physical equations, symmetry analysis, group classification, supersymmetry and supersymmetric equations, as well as the main work of this dissertation.
In Chapter 2, the main content and idea of AC = BD model theory and C-D pair, according to the mechanized construction of transformation of differential equations are presented.
In Chapter 3, based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality, mechanization, we improve the generalized Riccati equation rational expansion method and extend the Sub-ODE method. Furthermore, more exact solutions of nonlinear evolution equations with nonlinear terms of any order are obtained.
In Chapter 4, we solve the problem of the group classification of the nonlinear differential-difference equation of the form (?)_n = F_n(t,u_(n-1),u_n,u_(n+1)) invariant under Lie algebras through the method of the group classification of differential equations given by Zhdanov and Lahno.
In Chapter 5, for supersymmetric two bosons equations, we compute the adjoint representation of Lie superalgebra and conjugate classes of one-dimensional subalgebras. Then we find the preliminary classification of the group-invariant solutions and obtain rational solutions, exponential function solutions and trigonometric function solutions.
第一章介绍数学机械化、孤立子理论、数学物理方程的精确求解、对称分析、群分类、超对称和超对称方程的历史发展和研究现状,并介绍本文的选题及主要工作.
第二章介绍微分方程变换的机械化构造的AC=BD理论和C-D对理论的基本内容和思想.
第三章基于将非线性发展方程精确求解代数化、算法化、机械化的指导思想和AC=BD理论,改进广义Riccati方程有理展开法,并推广Sub-ODE方法,进而给出带有任意阶非线性项的非线性发展方程更多的精确解.
第四章利用Zhdanov和Lahno给出的求解偏微分方程群分类的方法研究非线性微分差分方程(?)_n=F_n(t,u_(n-1),u_n,u_(n+1))在李代数下不变的群分类.
第五章给出超对称二玻色子方程的李超代数的伴随表示关系及其在这种关系下一维子代数的共轭类,进一步计算群不变解的初步分类,并得到超对称二玻色子方程的指数函数解、三角函数解和有理解.
In this dissertation, some topics axe studied with the idea of mathematical mechanizationand AC=BD model, including exact solutions of nonlinear partial differential equations with nonlinear terms of any order, group classification of differential-difference equations and classification of the group-invariant solutions of supersymmetric equations.
In Chapter 1, we introduce the history and development of mathematical mechanization,soliton theory, solutions of the mathematical and physical equations, symmetry analysis, group classification, supersymmetry and supersymmetric equations, as well as the main work of this dissertation.
In Chapter 2, the main content and idea of AC = BD model theory and C-D pair, according to the mechanized construction of transformation of differential equations are presented.
In Chapter 3, based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality, mechanization, we improve the generalized Riccati equation rational expansion method and extend the Sub-ODE method. Furthermore, more exact solutions of nonlinear evolution equations with nonlinear terms of any order are obtained.
In Chapter 4, we solve the problem of the group classification of the nonlinear differential-difference equation of the form (?)_n = F_n(t,u_(n-1),u_n,u_(n+1)) invariant under Lie algebras through the method of the group classification of differential equations given by Zhdanov and Lahno.
In Chapter 5, for supersymmetric two bosons equations, we compute the adjoint representation of Lie superalgebra and conjugate classes of one-dimensional subalgebras. Then we find the preliminary classification of the group-invariant solutions and obtain rational solutions, exponential function solutions and trigonometric function solutions.
引文
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