孤立子与微分几何中某些问题的机械化方法
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摘要
本文以吴方法(吴代数消元法和吴微分消元法)为工具，研究了孤立子理论的某些问题、可积系统和微分几何中的部分定理。给出了求非线性演化方程精确解(孤子解、周期解、双周期解、有理函数解)的机械化方法；把吴微分特征列法和Reid的理论相结合应用于线性偏微分方程，计算解的规模；把吴微分特征列法应用于微分几何，给出部分定理的机械化证明。
第一章主要介绍了本文所涉及的概念，孤立子理论研究的起源和发展情况，孤立子与微分几何的关系以及国内、外学者在这些方面的工作和已经取得的成果。
第二章介绍了求解非线性偏微分方程的AC=BD模式及其应用。首先给出了C-D对和C-D可积系统的基本理论以及构造C-D对的方法。如何寻找变换是这一部分的重点内容。然后把AC=BD模式应用于微分几何，给出了微分几何中的C-D对和广义C-D可积系统。
第三章研究了齐次平衡法的改进和应用。把它应用于Boussinesq方程并和吴方法相结合，获得了许多新的孤子解和双周期解。把它应用于变系数KdV方程、DLW方程、SK方程、KK方程、KP方程，不仅得到了Bcklund变换，而且得到了更多的精确解。
第四章讨论了求非线性演化方程孤波解的若干方法：包括新的extended-tanh函数方法、扩展Riccati方程方法、射影Riccati方程方法、一般形式的Riccati方程方法，并给出了一般形式的Riccati方程多种形式的解。利用这些方法探讨了一类非线性演化方程，包括Burgers方程、广义Burgers-Fisher方程、Kuramoto-Sivashinsky方程的精确解(包括奇性孤波解，周期解和有理函数解)。进一步研究了高维变系数Burgers方程的类孤子解。在解决问题的过程中吴方法是最重要的基本工具。
第五章研究了非线性偏微分方程的雅可比椭圆函数解(双周期解)的机械化算法。首先提出了改进的Jacobi椭圆函数展开法，它是一种比sine-cosine方法和sn-cn函数法以及双曲函数法更有效更简单的方法。把它应用于组合KdV和mKdV方程，获得了许多雅可比椭圆函数解和其它精确解。然后，我们又提出了第一种和第二种椭圆方程法。特别给出了这两种椭圆方程更多形式的雅可比椭圆函数解，利用这些解，我们获得了一类非线性演化方程，包括耦合KdV方程、耦合mKdV方程的双周期解。在退化情况下，又得到其孤子解。最后，把它应用于高维变系数KP方程，获得了更多的双周期解。
第六章介绍了吴微分特征列法的基本理论及其应用。把它与Reid方法结合，应用于线性偏微分方程，得到了解的规模；把它应用于微分几何，得到微分几何中部分定理的机械化证明。
In this dissertation, we discuss some problems of soliton theory, the integrable systems and some theorems of differential geometry with the aid of Wu method (including Wu algebraic elimination method and Wu differential elimination theory). Some mechanical methods are presented to obtain exact solutions(including soliton solutions, periodic solutions, doubly-periodic solutions and rational solutions) of nonlinear evolution equations. Wu-Ritt differential characteristic set theory together with Reid method is applied to linear partial differential equations which possess physical significance to obtain the size of solutions. It is also applied to differential geometry to prove some theorems mechanically.
In Charter 1, we introduce some related definitions, the origin and development of soliton theory and the relations between soliton theory and differential geometry. The main works and achievements which have been obtained are presented.
Charter 2 is devoted to AC=BD model and its applications in partial differential equations and differential geometry. First, we give basic notations, basic theory of C-D pair and C-D integrable systems with the algorithm to construct C-D pair. How to seek transformation u = Cv is an important aspect in Charter 2. Then, AC=BD model is applied to differential geometry. C-D pair and general C-D integrable system are defined.
In Chapter 3, we study the applications of the improved homogenous balance method. We apply the method together with Wu algebraic elimination method to Boussinesq equation, and many new soliton and doubly-periodic solutions are obtained. Applying the method to SK, KK, KP, DLW equations and KdV equation with variable coefficients, we obtain not only their Backlund transformations but also new exact solutions.
Chapter 4 deals with some mechanical methods to obtain soliton solutions of nonlinear evolution equations including new extended-tanh function method, extended Ric-cati equation method, project Riccati equation method and generalized Riccati equation method. The multiple soliton solutions of the generalized Riccati equation are obtained. With these solutions, we obtain more exact solutions(including solitary wave solutions, periodic wave solutions and rational solutions) of a kind of nonlinear evolution equations, such as Burgers equation, general Burgers-Fisher equation and Kuramoto-Sivashinsky equation. More soliton-like solutions of the Burgers equation with variable coefficients are obtained by use of our method. Wu method is the most important basic tool during the course of solving these equations.
In Chapter 5, we present the mechanical methods to obtain doubly-periodic solutions
of nonlinear partial differential equations. First, we give the improved Jacobi elliptic function expansion method. It is more effective than the sine-cosine method, the sn-cn method and tanh-method. We apply the method to the combined KdV and mKdV equations to obtain Jacobi elliptic function solutions and other exact solutions. Then, we also present the first kind and the second kind of elliptic equation method, with which more Jacobi elliptic function solutions of these two kinds of equations are obtained. With these solutions, we obtain more doubly-periodic solutions of a class of nonlinear evolution equations, such as the coupled KdV and mKdV equations. These solutions are degenerated to soliton solutions under degenerated conditions. Finally, our method is applied to higher dimensional KP equation with variable coefficients to obtain more doubly-periodic solutions.
In the last chapter, we study basic theory of Wu-Ritt method and its applications. We apply Wu-Ritt method and Reid theory to linear partial differential equations to obtain the size of solutions. We also apply Wu-Ritt differential characteristic set to differential geometry to prove some theorems mechanically.
第一章主要介绍了本文所涉及的概念，孤立子理论研究的起源和发展情况，孤立子与微分几何的关系以及国内、外学者在这些方面的工作和已经取得的成果。
第二章介绍了求解非线性偏微分方程的AC=BD模式及其应用。首先给出了C-D对和C-D可积系统的基本理论以及构造C-D对的方法。如何寻找变换是这一部分的重点内容。然后把AC=BD模式应用于微分几何，给出了微分几何中的C-D对和广义C-D可积系统。
第三章研究了齐次平衡法的改进和应用。把它应用于Boussinesq方程并和吴方法相结合，获得了许多新的孤子解和双周期解。把它应用于变系数KdV方程、DLW方程、SK方程、KK方程、KP方程，不仅得到了Bcklund变换，而且得到了更多的精确解。
第四章讨论了求非线性演化方程孤波解的若干方法：包括新的extended-tanh函数方法、扩展Riccati方程方法、射影Riccati方程方法、一般形式的Riccati方程方法，并给出了一般形式的Riccati方程多种形式的解。利用这些方法探讨了一类非线性演化方程，包括Burgers方程、广义Burgers-Fisher方程、Kuramoto-Sivashinsky方程的精确解(包括奇性孤波解，周期解和有理函数解)。进一步研究了高维变系数Burgers方程的类孤子解。在解决问题的过程中吴方法是最重要的基本工具。
第五章研究了非线性偏微分方程的雅可比椭圆函数解(双周期解)的机械化算法。首先提出了改进的Jacobi椭圆函数展开法，它是一种比sine-cosine方法和sn-cn函数法以及双曲函数法更有效更简单的方法。把它应用于组合KdV和mKdV方程，获得了许多雅可比椭圆函数解和其它精确解。然后，我们又提出了第一种和第二种椭圆方程法。特别给出了这两种椭圆方程更多形式的雅可比椭圆函数解，利用这些解，我们获得了一类非线性演化方程，包括耦合KdV方程、耦合mKdV方程的双周期解。在退化情况下，又得到其孤子解。最后，把它应用于高维变系数KP方程，获得了更多的双周期解。
第六章介绍了吴微分特征列法的基本理论及其应用。把它与Reid方法结合，应用于线性偏微分方程，得到了解的规模；把它应用于微分几何，得到微分几何中部分定理的机械化证明。
In this dissertation, we discuss some problems of soliton theory, the integrable systems and some theorems of differential geometry with the aid of Wu method (including Wu algebraic elimination method and Wu differential elimination theory). Some mechanical methods are presented to obtain exact solutions(including soliton solutions, periodic solutions, doubly-periodic solutions and rational solutions) of nonlinear evolution equations. Wu-Ritt differential characteristic set theory together with Reid method is applied to linear partial differential equations which possess physical significance to obtain the size of solutions. It is also applied to differential geometry to prove some theorems mechanically.
In Charter 1, we introduce some related definitions, the origin and development of soliton theory and the relations between soliton theory and differential geometry. The main works and achievements which have been obtained are presented.
Charter 2 is devoted to AC=BD model and its applications in partial differential equations and differential geometry. First, we give basic notations, basic theory of C-D pair and C-D integrable systems with the algorithm to construct C-D pair. How to seek transformation u = Cv is an important aspect in Charter 2. Then, AC=BD model is applied to differential geometry. C-D pair and general C-D integrable system are defined.
In Chapter 3, we study the applications of the improved homogenous balance method. We apply the method together with Wu algebraic elimination method to Boussinesq equation, and many new soliton and doubly-periodic solutions are obtained. Applying the method to SK, KK, KP, DLW equations and KdV equation with variable coefficients, we obtain not only their Backlund transformations but also new exact solutions.
Chapter 4 deals with some mechanical methods to obtain soliton solutions of nonlinear evolution equations including new extended-tanh function method, extended Ric-cati equation method, project Riccati equation method and generalized Riccati equation method. The multiple soliton solutions of the generalized Riccati equation are obtained. With these solutions, we obtain more exact solutions(including solitary wave solutions, periodic wave solutions and rational solutions) of a kind of nonlinear evolution equations, such as Burgers equation, general Burgers-Fisher equation and Kuramoto-Sivashinsky equation. More soliton-like solutions of the Burgers equation with variable coefficients are obtained by use of our method. Wu method is the most important basic tool during the course of solving these equations.
In Chapter 5, we present the mechanical methods to obtain doubly-periodic solutions
of nonlinear partial differential equations. First, we give the improved Jacobi elliptic function expansion method. It is more effective than the sine-cosine method, the sn-cn method and tanh-method. We apply the method to the combined KdV and mKdV equations to obtain Jacobi elliptic function solutions and other exact solutions. Then, we also present the first kind and the second kind of elliptic equation method, with which more Jacobi elliptic function solutions of these two kinds of equations are obtained. With these solutions, we obtain more doubly-periodic solutions of a class of nonlinear evolution equations, such as the coupled KdV and mKdV equations. These solutions are degenerated to soliton solutions under degenerated conditions. Finally, our method is applied to higher dimensional KP equation with variable coefficients to obtain more doubly-periodic solutions.
In the last chapter, we study basic theory of Wu-Ritt method and its applications. We apply Wu-Ritt method and Reid theory to linear partial differential equations to obtain the size of solutions. We also apply Wu-Ritt differential characteristic set to differential geometry to prove some theorems mechanically.
引文
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