非线性发展方程求解法的研究与数学机械化实现
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摘要
本文根据数学机械化思想，以计算机符号和数值计算软件为工具，研究了孤立子理论中若干重要的非线性发展方程的求解方法及其相关问题，提出和发展了一系列求非线性发展方程解的方法，并在计算系统Maple或MATLAB上予以机械化实现。将数学机械化方法应用于相关学科，开发了数学机械化软件平台。主要的工作如下：
第一章介绍了孤立子理论和非线性发展方程求解理论及其数学机械化研究的历史发展和现状。同时介绍了一些关于这些学科的国内外学者所取得的成果。
第二章介绍了构造非线性发展方程精确解的“AC=BD”模式和构造“C-D”对的算法，利用Maple和“AC=BD+R”带余除法构造精确解的具体算法。
第三章基于将非线性发展方程求解统一化，算法化，机械化的思想，运用吴方法和符号计算的工具，建立了广义双曲函数的理论，提出了求非线性发展方程的广义双曲函数解和研究解的长时间行态及其相关问题的一系列方法。主要内容如下：
(1)给出广义双曲函数的定义和代数与微分性质及其证明，构造非线性发展方程解的广义双曲函数变换的定义和一些具体形式。
(2)提出了广义双曲函数-B(?)cklund变换方法，将其应用于解非线性发展方程组，求出了许多新的更一般的精确解。用计算机数值模拟方法研究了一些解的长时间的行态，结果表明这些新解具有良好的长时间的稳定性。
(3)提出了划分非线性发展方程的广义双曲函数解的长时间行态的三段法，并将其应用于研究一些非线性发展方程的广义双曲函数解的长时间稳定性，检验该方法的有效性。另外，还分别提出了修正广义双曲函数解和变系数解的长时间行态的方法。
(4)根据WTC方法和齐次平衡法构造B(?)cklund变换的方法的思想，提出了一种构造B(?)cklund变换的方法及其机械化算法，并将该方法应用于构造一些高阶高维的非线性发展方程的B(?)cklund变换，检验了有效性和可靠性。另外，还提出了与该方法相关的定理，并给出了证明。
(5)利用计算机数值模拟方法，广泛地研究了非线性发展方程的广义双曲函数解中的三个参数的不同取值对该解的局部性质和长时间行态的影响，一个非线性发展方程在同一种自-B(?)cklund变换下，取不同类型的种子解对该发展方程解的个数和解的形式的影响，不同类型的种子解对解的主部的影响，各种类型的广义双曲函数解的长时间行态，不同类型的非行波解和行波解的长时间行态的比较等问题，有一些新的发现，提出四个猜测。
第四章以符号计算软件Maple为工具，发展了构造非线性发展方程精确解的改进的F-展开法和推广的射影-Riccati方程法，提出了如下方法及其定理：
(1)构造了广义双曲函数-Riccati方程，提出了有关广义双曲函数-Riccati方程具有新的更一般的广义双曲函数解的定理、广义的射影Riccati方程和射影Riccati方程是广义双曲函数-Riccati方程的特例的定理，并且用Maple机械化方法给出了这两个定理的证明。
(2)利用广义双曲函数-Riccati方程，提出了广义双曲函数-Riccati方法，并用该方法求出了非线性发展方程的新的更一般形式的解。
(3)通过构造两类更一般的变换，提出了广义F-展开法和扩展的广义F-展开法。并将这些方法分别应用到一些非线性发展方程，结果成功地获得了这些方程的许多新的更一般的精确解。
第五章构造更一般的变换，给出类N孤子解的定义和猜测5，发展了Exp-函数方法，提出了Exp-B(?)cklund变换方法和Exp-类N孤子方法。利用这两种新方法获得了一些非线性发展方程的包含行波解和非行波解的更一般形式的精确解，并用计算机数值模拟方法研究了这类解的长时间行态。
第六章发展了求非线性发展方程的行波解的代数方法，提出了如下方法及其相关的定理：
(1)提出了一般形式的变换和相关定理，然后用Maple机械化方法证明了该定理。
(2)提出了求一阶任意次非线性常微分方程的精确解的机械化算法及其Maple程序，通过求六、八、十、十二次非线性常微分方程的某些一般形式的新的精确解，验证了该方法的有效性和可靠性。
(3)利用一阶任意次非线性常微分方程及其新的精确解，提出了广义的代数方法和扩展的广义的代数方法，并将它们分别应用到一些非线性发展方程，结果得到许多新的行波解和非行波解。
第七章改进了一些数值算法，提出了一类求非线性发展方程解的数值与解析混合运算的方法，求解常微分方程初值问题的改进的亚当斯方法等，提高了数值计算精度，并算法实现了机械化。另外，还提出了数值解、符号解、误差估计、输出结果图形可视化或表格化并举的设计数值计算机软件的新策略，开发了大量的数学计算机软件程序，建立了数值分析和高等数学的机械化软件MATLAB平台，使同类问题自动求解。
In this dissertation, under the guidance of mathematics mechanization and by means of computer algebraic and numerical systems, some problems in the theory of solitons are discussed and some methods for constructing the solutions of nonlinear evolution equations are presented. The methods presented are realized on the algebraic computation software Maple or MATLAB. The method of mathematics mechanization is applied to the related subjects and the operating systems of mathematics mechanization using MATLAB are set up. The description is as follows:
Chapter 1 of this dissertation is devoted to reviewing the history and development of soliton theory, some methods for seeking exact solutions of nonlinear evolution equations and the mathematics mechanization, with an emphasis on some achievements on the subjects involved in this dissertation.
Chapter 2 concerns the construction of exact solutions of nonlinear evolution equations under via the "AC=BD" theory introduced by Prof. H. Q. Zhang. The construction of the operators of C and D and some concrete algorithms for constructing the exact solution based on the division with remainder "AC=BD+R" in terms of the symbolic computation software Maple are introduced.
In chapter 3, based on the ideas of unification methods, algorithm realization and mechanization for solving nonlinear evolution equations, we found a new theory of generalized hyperbolic functions and present some new methods to construct the solutions of nonlinear evolution equations. Its main contents are as follows:
(1) A new definition of generalized hyperbolic functions, generalized hyperbolic function transformation and their properties are presented. Then some concrete formulae of generalized hyperbolic function transformation are also given for constructing the solutions of nonlinear evolution equations.
(2) A new generalized hyperbolic function - B(?)cklund transformation method is presented and applied to construct the exact solutions of some nonlinear evolution equations. As a result, a lot of new exact solutions in more general forms are obtained. Then we investigate the long-playing traveling state of the solutions via computer simulation and find that the solutions are of long time stability.
(3) A new method of partitioned long-playing traveling state of the generalized hyperbolic function solution of nonlinear evolution equations is presented. The validity of the method is tested by its application to investigating the long time stability of the solutions. In addition, two methods which revise long-playing traveling state of the generalized hyperbolic function solutions and the variable coefficient solutions of nonlinear evolution equations are given.
(4) Based on the ideas of the WTC method and the homogeneous balance method for constructing B(?)cklund transformation, a new method and its mechanization algorithm are suggested for constructing the B(?)cklund transformation. The validity and reliability of the method are tested by its application to two nonlinear evolution equatious of higher order and higher dimension. In addition, a new theorem about the method is presented and proved.
(5) We investigate in a wide context whether the different values of three parameters in the gener- alized hyperbolic function solution affect the local properties of the long-playing travelling state of the solutions, whether the different types of a seed solution of a nonlinear evolution equation under the same B(?)cklund transformation affect the shape and number of solitons, the seed solution affect on the main part of solution, long-playing travelling state of the variable (constant) coefficient non -travelling wave solution differs from travelling wave solution, and the long-playing travelling states of several types of generalized hyperbolic function solutions and so on by means of computer simulation. As a result, we find some new phenomena and suggest four guesses.
In chapter 4, by use of the symbolic computation software Maple, the general projective Riccati equation method and the improved F- expansion method are improved, and the resulting new methods and theorems are devised to construct the exact solutions for nonlinear evolution equations.
(1) A new generalized hyperbolic function-Riccati equation is constructed and two theorems, which shows that the equation possesses new and more general generalized hyperbolic function solution including the projective Riccati equation and general projective Riccati equation, are devised and proved by a mechanization method using Maple.
(2) By means of the generalized hyperbolic function-Riccati equation, a new generalized hyperbolic function -Riccati method is presented and applied to some nonlinear evolution equations. As a result, new exact solutions in more general forms are obtained.
(3) By devising two kinds of new and more general transformation, a new generalized F-expansion method and a new extended generalized F- expansion method are proposed for constructing the exact solutions of nonlinear evolution equations. These new methods are applied to some nonlinear evolution equations and more general exact solutions are obtained.
In chapter 5, a new definition of N soliton-like solution, guess 5, and a new and more general transformation are given, the Exp- function method is developed, and a new Exp -B(?)cklund method and a new Exp -N soliton-tike method are presented. Some more general exact solutions including nontravelling wave solutions and traveling wave solutions of some nonlinear evolution equations are derived using these methods. The long-playing traveling state of the solutions is sought by means of computer simulation.
In chapter 6, the algebraic methods for constructing the traveling wave solutions of nonlinear evolution equations are developed and the following new methods and theorems are presented.
(1) A new and general transformation, and a new theorem which is proved in Maple are presented.
(2) A new mechanization method to find the exact solutions of a first-order nonlinear ordinary differential equation with any degree is presented. The validity and reliability of the method are tested by its application to the first-order nonlinear ordinary differential equation with six degree, eight degree, ten degree, and twelve degree.
(3) A new generalized algebraic method, an extended generalized algebraic method, and their algorithms are suggested based on a nonlinear ordinary differential equation with any degree. Some nonlinear evolution equations are chosen to illustrate our algorithm so that more families of new exact solutions are obtained, which contain both non-traveling and traveling wave solutions.
In chapter 7, some numerical methods are improved, a new compound operation method consisting of numerical and analytical methods to solve nonlinear evolution equations and the improved Adams method are proposed, and the accuracy of numerical computation is raised. A new tactics of devising a new system for handling numerical computation orders, which can actualize synchronously that numerical solutions, sign solution, and their error estimates are outputted in a figure format or a table format, is presented. A great deal of the software on mathematics mechanization is developed and two systems for handling the computation and the figures orders on numerical analysis and advanced mathematics using MATLAB are set up so that the same type of problem is solved automatically.
第一章介绍了孤立子理论和非线性发展方程求解理论及其数学机械化研究的历史发展和现状。同时介绍了一些关于这些学科的国内外学者所取得的成果。
第二章介绍了构造非线性发展方程精确解的“AC=BD”模式和构造“C-D”对的算法，利用Maple和“AC=BD+R”带余除法构造精确解的具体算法。
第三章基于将非线性发展方程求解统一化，算法化，机械化的思想，运用吴方法和符号计算的工具，建立了广义双曲函数的理论，提出了求非线性发展方程的广义双曲函数解和研究解的长时间行态及其相关问题的一系列方法。主要内容如下：
(1)给出广义双曲函数的定义和代数与微分性质及其证明，构造非线性发展方程解的广义双曲函数变换的定义和一些具体形式。
(2)提出了广义双曲函数-B(?)cklund变换方法，将其应用于解非线性发展方程组，求出了许多新的更一般的精确解。用计算机数值模拟方法研究了一些解的长时间的行态，结果表明这些新解具有良好的长时间的稳定性。
(3)提出了划分非线性发展方程的广义双曲函数解的长时间行态的三段法，并将其应用于研究一些非线性发展方程的广义双曲函数解的长时间稳定性，检验该方法的有效性。另外，还分别提出了修正广义双曲函数解和变系数解的长时间行态的方法。
(4)根据WTC方法和齐次平衡法构造B(?)cklund变换的方法的思想，提出了一种构造B(?)cklund变换的方法及其机械化算法，并将该方法应用于构造一些高阶高维的非线性发展方程的B(?)cklund变换，检验了有效性和可靠性。另外，还提出了与该方法相关的定理，并给出了证明。
(5)利用计算机数值模拟方法，广泛地研究了非线性发展方程的广义双曲函数解中的三个参数的不同取值对该解的局部性质和长时间行态的影响，一个非线性发展方程在同一种自-B(?)cklund变换下，取不同类型的种子解对该发展方程解的个数和解的形式的影响，不同类型的种子解对解的主部的影响，各种类型的广义双曲函数解的长时间行态，不同类型的非行波解和行波解的长时间行态的比较等问题，有一些新的发现，提出四个猜测。
第四章以符号计算软件Maple为工具，发展了构造非线性发展方程精确解的改进的F-展开法和推广的射影-Riccati方程法，提出了如下方法及其定理：
(1)构造了广义双曲函数-Riccati方程，提出了有关广义双曲函数-Riccati方程具有新的更一般的广义双曲函数解的定理、广义的射影Riccati方程和射影Riccati方程是广义双曲函数-Riccati方程的特例的定理，并且用Maple机械化方法给出了这两个定理的证明。
(2)利用广义双曲函数-Riccati方程，提出了广义双曲函数-Riccati方法，并用该方法求出了非线性发展方程的新的更一般形式的解。
(3)通过构造两类更一般的变换，提出了广义F-展开法和扩展的广义F-展开法。并将这些方法分别应用到一些非线性发展方程，结果成功地获得了这些方程的许多新的更一般的精确解。
第五章构造更一般的变换，给出类N孤子解的定义和猜测5，发展了Exp-函数方法，提出了Exp-B(?)cklund变换方法和Exp-类N孤子方法。利用这两种新方法获得了一些非线性发展方程的包含行波解和非行波解的更一般形式的精确解，并用计算机数值模拟方法研究了这类解的长时间行态。
第六章发展了求非线性发展方程的行波解的代数方法，提出了如下方法及其相关的定理：
(1)提出了一般形式的变换和相关定理，然后用Maple机械化方法证明了该定理。
(2)提出了求一阶任意次非线性常微分方程的精确解的机械化算法及其Maple程序，通过求六、八、十、十二次非线性常微分方程的某些一般形式的新的精确解，验证了该方法的有效性和可靠性。
(3)利用一阶任意次非线性常微分方程及其新的精确解，提出了广义的代数方法和扩展的广义的代数方法，并将它们分别应用到一些非线性发展方程，结果得到许多新的行波解和非行波解。
第七章改进了一些数值算法，提出了一类求非线性发展方程解的数值与解析混合运算的方法，求解常微分方程初值问题的改进的亚当斯方法等，提高了数值计算精度，并算法实现了机械化。另外，还提出了数值解、符号解、误差估计、输出结果图形可视化或表格化并举的设计数值计算机软件的新策略，开发了大量的数学计算机软件程序，建立了数值分析和高等数学的机械化软件MATLAB平台，使同类问题自动求解。
In this dissertation, under the guidance of mathematics mechanization and by means of computer algebraic and numerical systems, some problems in the theory of solitons are discussed and some methods for constructing the solutions of nonlinear evolution equations are presented. The methods presented are realized on the algebraic computation software Maple or MATLAB. The method of mathematics mechanization is applied to the related subjects and the operating systems of mathematics mechanization using MATLAB are set up. The description is as follows:
Chapter 1 of this dissertation is devoted to reviewing the history and development of soliton theory, some methods for seeking exact solutions of nonlinear evolution equations and the mathematics mechanization, with an emphasis on some achievements on the subjects involved in this dissertation.
Chapter 2 concerns the construction of exact solutions of nonlinear evolution equations under via the "AC=BD" theory introduced by Prof. H. Q. Zhang. The construction of the operators of C and D and some concrete algorithms for constructing the exact solution based on the division with remainder "AC=BD+R" in terms of the symbolic computation software Maple are introduced.
In chapter 3, based on the ideas of unification methods, algorithm realization and mechanization for solving nonlinear evolution equations, we found a new theory of generalized hyperbolic functions and present some new methods to construct the solutions of nonlinear evolution equations. Its main contents are as follows:
(1) A new definition of generalized hyperbolic functions, generalized hyperbolic function transformation and their properties are presented. Then some concrete formulae of generalized hyperbolic function transformation are also given for constructing the solutions of nonlinear evolution equations.
(2) A new generalized hyperbolic function - B(?)cklund transformation method is presented and applied to construct the exact solutions of some nonlinear evolution equations. As a result, a lot of new exact solutions in more general forms are obtained. Then we investigate the long-playing traveling state of the solutions via computer simulation and find that the solutions are of long time stability.
(3) A new method of partitioned long-playing traveling state of the generalized hyperbolic function solution of nonlinear evolution equations is presented. The validity of the method is tested by its application to investigating the long time stability of the solutions. In addition, two methods which revise long-playing traveling state of the generalized hyperbolic function solutions and the variable coefficient solutions of nonlinear evolution equations are given.
(4) Based on the ideas of the WTC method and the homogeneous balance method for constructing B(?)cklund transformation, a new method and its mechanization algorithm are suggested for constructing the B(?)cklund transformation. The validity and reliability of the method are tested by its application to two nonlinear evolution equatious of higher order and higher dimension. In addition, a new theorem about the method is presented and proved.
(5) We investigate in a wide context whether the different values of three parameters in the gener- alized hyperbolic function solution affect the local properties of the long-playing travelling state of the solutions, whether the different types of a seed solution of a nonlinear evolution equation under the same B(?)cklund transformation affect the shape and number of solitons, the seed solution affect on the main part of solution, long-playing travelling state of the variable (constant) coefficient non -travelling wave solution differs from travelling wave solution, and the long-playing travelling states of several types of generalized hyperbolic function solutions and so on by means of computer simulation. As a result, we find some new phenomena and suggest four guesses.
In chapter 4, by use of the symbolic computation software Maple, the general projective Riccati equation method and the improved F- expansion method are improved, and the resulting new methods and theorems are devised to construct the exact solutions for nonlinear evolution equations.
(1) A new generalized hyperbolic function-Riccati equation is constructed and two theorems, which shows that the equation possesses new and more general generalized hyperbolic function solution including the projective Riccati equation and general projective Riccati equation, are devised and proved by a mechanization method using Maple.
(2) By means of the generalized hyperbolic function-Riccati equation, a new generalized hyperbolic function -Riccati method is presented and applied to some nonlinear evolution equations. As a result, new exact solutions in more general forms are obtained.
(3) By devising two kinds of new and more general transformation, a new generalized F-expansion method and a new extended generalized F- expansion method are proposed for constructing the exact solutions of nonlinear evolution equations. These new methods are applied to some nonlinear evolution equations and more general exact solutions are obtained.
In chapter 5, a new definition of N soliton-like solution, guess 5, and a new and more general transformation are given, the Exp- function method is developed, and a new Exp -B(?)cklund method and a new Exp -N soliton-tike method are presented. Some more general exact solutions including nontravelling wave solutions and traveling wave solutions of some nonlinear evolution equations are derived using these methods. The long-playing traveling state of the solutions is sought by means of computer simulation.
In chapter 6, the algebraic methods for constructing the traveling wave solutions of nonlinear evolution equations are developed and the following new methods and theorems are presented.
(1) A new and general transformation, and a new theorem which is proved in Maple are presented.
(2) A new mechanization method to find the exact solutions of a first-order nonlinear ordinary differential equation with any degree is presented. The validity and reliability of the method are tested by its application to the first-order nonlinear ordinary differential equation with six degree, eight degree, ten degree, and twelve degree.
(3) A new generalized algebraic method, an extended generalized algebraic method, and their algorithms are suggested based on a nonlinear ordinary differential equation with any degree. Some nonlinear evolution equations are chosen to illustrate our algorithm so that more families of new exact solutions are obtained, which contain both non-traveling and traveling wave solutions.
In chapter 7, some numerical methods are improved, a new compound operation method consisting of numerical and analytical methods to solve nonlinear evolution equations and the improved Adams method are proposed, and the accuracy of numerical computation is raised. A new tactics of devising a new system for handling numerical computation orders, which can actualize synchronously that numerical solutions, sign solution, and their error estimates are outputted in a figure format or a table format, is presented. A great deal of the software on mathematics mechanization is developed and two systems for handling the computation and the figures orders on numerical analysis and advanced mathematics using MATLAB are set up so that the same type of problem is solved automatically.
引文
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