论非线性发展方程求解中辅助方程法的历史演进
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摘要
1834年8月,英国科学家罗素发现了孤立波自然现象.1895年,荷兰阿姆斯特丹大学的数学家德弗里斯(G.de Vries)在导师柯特维格(D.J.Korteweg)的指导下,研究单方向运动的浅水波时,建立了描述罗素孤立波现象的数学模型KdV方程,从理论上肯定了孤立波解的存在性.1955年,美国物理学家费米(Enrico Fermi),帕斯塔(John Pasta)和犹拉姆(Stan Ulam)提出的著名的FPU问题,对于发现孤立子提供了第一个实验依据.1965年,美国Princeton大学应用数学家扎布斯基(N.J.Zabusky)和实验室的克鲁斯卡尔(M.D.Kruskal)发现了FPU问题中弦的位移满足KdV方程,而且他们通过计算机模拟重现了孤立波相互作用时表现出类此于粒子的性质,并由此提出“孤立子”的概念.孤立子概念的提出证明了孤立波解的稳定性.
最近50多年来,人们利用计算机技术,在非线性光学中发现光孤子并应用于通信领域取得了成功.生物学中发现了达维多夫(Davydov)孤立子,海洋学中发现了内孤立波.另外,在凝聚态物理、激光物理、超导物理、经济学、人口问题和医学等诸多科学领域中相继发现了光滑孤立子解、尖峰孤立子解和紧孤立子解等多种孤立子.
孤立子理论的研究内容大致分为以下两类.
(1)构造系统的求解方法：即构造和发展求解非线性方程的一种系统的方法.这里指的非线性方程包括非线性偏微分方程,非线性常微分方程,非线性积分微分方程和非线性差分微分方程.对于许多非线性发展方程,已经有了多种有效的求解方法,但是没有一种通用的方法.
(2)解释解的性质：研究解释可积方程的代数和几何的一系列美妙的性质.这里所说的可积方程是能够转化成线性方程的非线性方程.对于研究解的性质方面一般有如下三个情况.第一种情况：当难以获得显示精确解时,分析研究非线性发展方程的适定性问题；第二种情况：利用计算数学的理论知识和计算机,对解进行模拟分析研究；第三种情况：利用试探法和构造变换法等数学技巧,获得非线性发展方程的精确解.虽然以上三种研究方法的角度不同,但是目的都是解释解的变化规律.
数学史研究数学概念、数学方法和数学思想的起源与发展,以及与社会政治、经济和一般的文化的联系.1974年,吴文俊开始研究中国数学史.他在“古证复原”原则下,利用“反辉格”与“中西方数学对比”相结合的综合性方法来研究中国传统数学,揭开了中国数学的构造性和机械化性两个特点.在此基础上与计算机技术相结合发明了著名的“吴消元法”.吴文俊的工作成就是“古为今用”的典范.他提出的“新方法论”对于数学史和数学研究工作来说具有指导性和启发性作用.
构造非线性发展方程的精确解是孤立子理论的重要研究课题之一.试探函数法与辅助方程法在构造非线性发展方程精确解领域发挥了非常重要的作用,已经获得了许多新成果.本文从“吴消元法”的发明得到启示,利用“新方法论”对2009年以前的辅助方程法和试探函数法有关的大量文献进行认真比较和仔细分析研究,获得了这两种方法的构造性和机械化性.在第四章中总结了试探函数法的构造性和机械化性两大特点.在此基础上,提出了新的试探函数法,构造了非线性连续(离散)发展方程新的精确解.
在第五章中首先通过对Riccati方程法等辅助方程法有关的大量文献进行研究,梳理了辅助方程法的思想基础和来源问题,总结了辅助方程法的四个应用步骤体现了该方法的构造性和机械化性两大特点.在此基础上,初步发挥辅助方程法的两大特点,提出了三角函数型辅助方程法与双曲函数型辅助方程法等新的方法,构造了非线性发展方程的新精确解.
(1)把非线性发展方程转化为非线性常微分方程的变换具有构造性.
(2)辅助方程与非线性常微分方程的形式解具有构造性.
(3)非线性方程组的求解问题具有机械化性.
(4)非线性发展方程解的验证具有机械化性.
理论上说：《非线性发展方程存在无穷多个解》.但是,辅助方程法有关的诸多博士(硕士)学位论文以及相关的文献只获得了有限多个精确解.本文为了获得非线性发展方程的无穷序列精确解,挖掘辅助方程法的两大特点的含义获得了Riccati方程、第一种椭圆辅助方程、第二种椭圆辅助方程等几种常用辅助方程的自Backlund变换、拟Backlund变换和解的非线性叠加公式,构造了连续(离散)和变系数(常系数)非线性发展方程的多种类型的无穷序列新精确解.
(1)单函数型无穷序列精确解.就是Jacobi椭圆函数、双曲函数、三角函数和有理函数单独构成的无穷序列新精确解.这里包括无穷序列光滑孤立波解、无穷序列尖峰孤立波解和尤穷序列紧孤立子解.本文不仅获得了K(m,n)方程、Degasperis-Procesi方程和CH方程的无穷序列尖峰孤立波解和无穷序列紧孤立子解,而目.其他的非线性发展方程中也获得了此类精确解.
(2)复合函数型无穷序列精确解.就是Jacobi椭圆函数、双曲函数、三角函数和有理函数通过几种形式复合而成的无穷序列精确解.这里包括光滑孤立波解、尖峰孤立波解和紧孤立子解通过几种形式复合而成的无穷序列新精确解.
In 1834, J. Scott Russell observed natural phenomenon of solitary waves. In 1895, D.J.Korteweg and G.de Vries governed mathematical model for descriptirrg phenomenon of Rus-sell solitary wave in KdV equation when they investigated shallow water wave in one dimensional, and confirmed the existence of solitary wave solutions in theory. In 1955, Enrico Fermi, John Pasta and Stan Ulam presented FPU problem which proposed the first experimental foundation for discoverying soliton.
In 1965, N.J.Zabusky and M.D.Kruskal discovered the length between speings in FPU problem satisfying KdV equation and reappeared the nature of the particles as interaction of solitary wave by computer simulation. Because of this, Zabusky and Kruskal named these special waves as solitons, which testified stabilization of solitary wave solutions.
In the rencent fifty years, with the development of the computer, optical soliton, Davydov soliton and inner solitary wave were discovered in the nonlinear optics, biology and oceanol-ogy, respectively. And smooth soliton solutions, peak soliton solutions and compact soliton solutions, etc in several fields are investigated, such as condense state physics, laser physics, superconductivity physics, economics, population problem, medical science and so on.
Now we would like to extend the two contents of the study of solitary theory as follows
(1) Constructing solving method of systems. The systematic method for constructing and developing solving nonlinear equations was presented, which include nonlinear partial differential equations, nonlinear ordinary differential equations, nonlinear integrable differential equations and nonlinear difference differential equations. Many powerful solving methods of nonlinear evolution equations are introduced, but a few common methods are proposed.
(2) Explaining the property of solutions. One of our aim is to study and explain many more algebraic and geometric properties of integrable equations, where integrable equations are nonlinear equations which can transformed into linear equations. Here will give three aspects about properties of solutions. For example, firstly, we will analyze and study qualitative prob-lems of nonlinear evolution equations when explicit exact solutions are difficult to be obtained. Secondly, the solutions are simulated and analyzed with the help of computation mathematical theory and computer. Finally, exact solutions of nonlinear evolution equations are obtained using trial function methods and constructive transformation methods, etc. Though the three research methods are different, the aims are to explain change rule of solutions.
The history of mathematics is to study origin and development of mathematics concepts, mathematics method and mathematics thinking, and the relation between these with social politics, economics and general cultures. In 1974, Wu Wentsun began to study the history of Chinese mathematics. Under "Make the Past to the Present" principle, he applied the method for combing anti-Whig with the comparison of Chinese and Western mathematics to study the Chinese classical mathematics and obtained the two characteristics of constructively and mechanization about Chinese mathematics. On this basis with computer, the world-famous "Wu elimination method" was presented. Therefore, the work of Wu Wentsun is a model of make the past to the present, and new methodology Wu proposed is the guidance and enlightening for the history and the research of mathematics.
The construction of exact solutions to nonlinear evolution equations is one of important researches in the solitary theory. The auxiliary method and trial function method have played significance on constructing exact solutions of nonlinear evolution equations, and much more work has been done. In the dissertation, according to Wu elimination method and applying new methodology to study references on the auxiliary equation method and the trial function method before 2009, we have searched for the two characteristics of constructively and mechanization of two methods. Hence, in chapter 4, we will summarize the two characteristics of constructively and mechanization about, the trial function method and propose the new trial function method to seek new exact solutions of nonlinear continuous(or discrete) evolution equations.
In chapter 5, based on the references, ideological foundation and sources of Riccati equation and other auxiliary equation methods, we sum up four applicable procedures to reflect the two characteristics of constructively and mechanization about auxiliary equation methods. Accord-ing to this, developing the characteristics of auxiliary equation method, the auxiliary equation methods of triangular function type and hyperbolic function type,etc. have been presented to construct exact solutions of nonlinear evolution equations.
(1) There is constructive for becoming nonlinear envolution equations into nonlinear ordi-nary differential equations.
(2) There is constructive for formal solutions of auxiliary equations and nonlinear ordinary differential equations.
(3) There is mechanization for solving the sets of nonlinear equations.
(4) There is mechanization for illustrating the solutions of nonlinear evolution equations.
In theory, nonlinear envolution equations exist infinite solutions. However, the auxiliary equation methods in many Master's thesises, doctoral dissertations and references can obtain fi-nite exact solutions. Therefore, in this dissertation, to seek new infinite sequence exact solutions to nonlinear evolution equations, studying highly references and summing up the characteris-tics of the auxiliary equation method, the auto-Backlund transformation, the quasi-Backlund transformation and the formula of nonlinear superposition of the solutions of Riccati equation, the first kind of elliptic equation and the second kind of elliptic equation,etc. are presented to construct new infinite sequence exact solutions of continuous(or discrete) nonlinear envolution equations with variable coefficients(or constant coefficients) as follows.
Firstly, infinite sequences exact solutions of single function. The infinite sequences exact solutions are constructed by Jacobi elliptic function, hyperbolic function, trianglar function and rational function. respectively, which include infinite sequences smooth solitary wave solutions, infinite sequences peak solitary wave solutions and infinite sequences compact soliton solutions. In this dissertation, beside K(m,n) equation, Degasperis-Procesi equation and CH equation, infinite sequences peak solitary wave solutions and infinite sequences compact soliton solutions are discovered in other nonlinear evolution equations.
Secondly. infinite sequences exact solutions of composite function.The infinite sequences exact solutions are composited by Jacobi elliptic function, hyperbolic function, trianglar func- tion and rational function in several forms, where include infinite sequences exact solutions are combined by smooth solitary wave solutions, peak solitary wave solutions and compact soliton solutions.
最近50多年来,人们利用计算机技术,在非线性光学中发现光孤子并应用于通信领域取得了成功.生物学中发现了达维多夫(Davydov)孤立子,海洋学中发现了内孤立波.另外,在凝聚态物理、激光物理、超导物理、经济学、人口问题和医学等诸多科学领域中相继发现了光滑孤立子解、尖峰孤立子解和紧孤立子解等多种孤立子.
孤立子理论的研究内容大致分为以下两类.
(1)构造系统的求解方法：即构造和发展求解非线性方程的一种系统的方法.这里指的非线性方程包括非线性偏微分方程,非线性常微分方程,非线性积分微分方程和非线性差分微分方程.对于许多非线性发展方程,已经有了多种有效的求解方法,但是没有一种通用的方法.
(2)解释解的性质：研究解释可积方程的代数和几何的一系列美妙的性质.这里所说的可积方程是能够转化成线性方程的非线性方程.对于研究解的性质方面一般有如下三个情况.第一种情况：当难以获得显示精确解时,分析研究非线性发展方程的适定性问题；第二种情况：利用计算数学的理论知识和计算机,对解进行模拟分析研究；第三种情况：利用试探法和构造变换法等数学技巧,获得非线性发展方程的精确解.虽然以上三种研究方法的角度不同,但是目的都是解释解的变化规律.
数学史研究数学概念、数学方法和数学思想的起源与发展,以及与社会政治、经济和一般的文化的联系.1974年,吴文俊开始研究中国数学史.他在“古证复原”原则下,利用“反辉格”与“中西方数学对比”相结合的综合性方法来研究中国传统数学,揭开了中国数学的构造性和机械化性两个特点.在此基础上与计算机技术相结合发明了著名的“吴消元法”.吴文俊的工作成就是“古为今用”的典范.他提出的“新方法论”对于数学史和数学研究工作来说具有指导性和启发性作用.
构造非线性发展方程的精确解是孤立子理论的重要研究课题之一.试探函数法与辅助方程法在构造非线性发展方程精确解领域发挥了非常重要的作用,已经获得了许多新成果.本文从“吴消元法”的发明得到启示,利用“新方法论”对2009年以前的辅助方程法和试探函数法有关的大量文献进行认真比较和仔细分析研究,获得了这两种方法的构造性和机械化性.在第四章中总结了试探函数法的构造性和机械化性两大特点.在此基础上,提出了新的试探函数法,构造了非线性连续(离散)发展方程新的精确解.
在第五章中首先通过对Riccati方程法等辅助方程法有关的大量文献进行研究,梳理了辅助方程法的思想基础和来源问题,总结了辅助方程法的四个应用步骤体现了该方法的构造性和机械化性两大特点.在此基础上,初步发挥辅助方程法的两大特点,提出了三角函数型辅助方程法与双曲函数型辅助方程法等新的方法,构造了非线性发展方程的新精确解.
(1)把非线性发展方程转化为非线性常微分方程的变换具有构造性.
(2)辅助方程与非线性常微分方程的形式解具有构造性.
(3)非线性方程组的求解问题具有机械化性.
(4)非线性发展方程解的验证具有机械化性.
理论上说：《非线性发展方程存在无穷多个解》.但是,辅助方程法有关的诸多博士(硕士)学位论文以及相关的文献只获得了有限多个精确解.本文为了获得非线性发展方程的无穷序列精确解,挖掘辅助方程法的两大特点的含义获得了Riccati方程、第一种椭圆辅助方程、第二种椭圆辅助方程等几种常用辅助方程的自Backlund变换、拟Backlund变换和解的非线性叠加公式,构造了连续(离散)和变系数(常系数)非线性发展方程的多种类型的无穷序列新精确解.
(1)单函数型无穷序列精确解.就是Jacobi椭圆函数、双曲函数、三角函数和有理函数单独构成的无穷序列新精确解.这里包括无穷序列光滑孤立波解、无穷序列尖峰孤立波解和尤穷序列紧孤立子解.本文不仅获得了K(m,n)方程、Degasperis-Procesi方程和CH方程的无穷序列尖峰孤立波解和无穷序列紧孤立子解,而目.其他的非线性发展方程中也获得了此类精确解.
(2)复合函数型无穷序列精确解.就是Jacobi椭圆函数、双曲函数、三角函数和有理函数通过几种形式复合而成的无穷序列精确解.这里包括光滑孤立波解、尖峰孤立波解和紧孤立子解通过几种形式复合而成的无穷序列新精确解.
In 1834, J. Scott Russell observed natural phenomenon of solitary waves. In 1895, D.J.Korteweg and G.de Vries governed mathematical model for descriptirrg phenomenon of Rus-sell solitary wave in KdV equation when they investigated shallow water wave in one dimensional, and confirmed the existence of solitary wave solutions in theory. In 1955, Enrico Fermi, John Pasta and Stan Ulam presented FPU problem which proposed the first experimental foundation for discoverying soliton.
In 1965, N.J.Zabusky and M.D.Kruskal discovered the length between speings in FPU problem satisfying KdV equation and reappeared the nature of the particles as interaction of solitary wave by computer simulation. Because of this, Zabusky and Kruskal named these special waves as solitons, which testified stabilization of solitary wave solutions.
In the rencent fifty years, with the development of the computer, optical soliton, Davydov soliton and inner solitary wave were discovered in the nonlinear optics, biology and oceanol-ogy, respectively. And smooth soliton solutions, peak soliton solutions and compact soliton solutions, etc in several fields are investigated, such as condense state physics, laser physics, superconductivity physics, economics, population problem, medical science and so on.
Now we would like to extend the two contents of the study of solitary theory as follows
(1) Constructing solving method of systems. The systematic method for constructing and developing solving nonlinear equations was presented, which include nonlinear partial differential equations, nonlinear ordinary differential equations, nonlinear integrable differential equations and nonlinear difference differential equations. Many powerful solving methods of nonlinear evolution equations are introduced, but a few common methods are proposed.
(2) Explaining the property of solutions. One of our aim is to study and explain many more algebraic and geometric properties of integrable equations, where integrable equations are nonlinear equations which can transformed into linear equations. Here will give three aspects about properties of solutions. For example, firstly, we will analyze and study qualitative prob-lems of nonlinear evolution equations when explicit exact solutions are difficult to be obtained. Secondly, the solutions are simulated and analyzed with the help of computation mathematical theory and computer. Finally, exact solutions of nonlinear evolution equations are obtained using trial function methods and constructive transformation methods, etc. Though the three research methods are different, the aims are to explain change rule of solutions.
The history of mathematics is to study origin and development of mathematics concepts, mathematics method and mathematics thinking, and the relation between these with social politics, economics and general cultures. In 1974, Wu Wentsun began to study the history of Chinese mathematics. Under "Make the Past to the Present" principle, he applied the method for combing anti-Whig with the comparison of Chinese and Western mathematics to study the Chinese classical mathematics and obtained the two characteristics of constructively and mechanization about Chinese mathematics. On this basis with computer, the world-famous "Wu elimination method" was presented. Therefore, the work of Wu Wentsun is a model of make the past to the present, and new methodology Wu proposed is the guidance and enlightening for the history and the research of mathematics.
The construction of exact solutions to nonlinear evolution equations is one of important researches in the solitary theory. The auxiliary method and trial function method have played significance on constructing exact solutions of nonlinear evolution equations, and much more work has been done. In the dissertation, according to Wu elimination method and applying new methodology to study references on the auxiliary equation method and the trial function method before 2009, we have searched for the two characteristics of constructively and mechanization of two methods. Hence, in chapter 4, we will summarize the two characteristics of constructively and mechanization about, the trial function method and propose the new trial function method to seek new exact solutions of nonlinear continuous(or discrete) evolution equations.
In chapter 5, based on the references, ideological foundation and sources of Riccati equation and other auxiliary equation methods, we sum up four applicable procedures to reflect the two characteristics of constructively and mechanization about auxiliary equation methods. Accord-ing to this, developing the characteristics of auxiliary equation method, the auxiliary equation methods of triangular function type and hyperbolic function type,etc. have been presented to construct exact solutions of nonlinear evolution equations.
(1) There is constructive for becoming nonlinear envolution equations into nonlinear ordi-nary differential equations.
(2) There is constructive for formal solutions of auxiliary equations and nonlinear ordinary differential equations.
(3) There is mechanization for solving the sets of nonlinear equations.
(4) There is mechanization for illustrating the solutions of nonlinear evolution equations.
In theory, nonlinear envolution equations exist infinite solutions. However, the auxiliary equation methods in many Master's thesises, doctoral dissertations and references can obtain fi-nite exact solutions. Therefore, in this dissertation, to seek new infinite sequence exact solutions to nonlinear evolution equations, studying highly references and summing up the characteris-tics of the auxiliary equation method, the auto-Backlund transformation, the quasi-Backlund transformation and the formula of nonlinear superposition of the solutions of Riccati equation, the first kind of elliptic equation and the second kind of elliptic equation,etc. are presented to construct new infinite sequence exact solutions of continuous(or discrete) nonlinear envolution equations with variable coefficients(or constant coefficients) as follows.
Firstly, infinite sequences exact solutions of single function. The infinite sequences exact solutions are constructed by Jacobi elliptic function, hyperbolic function, trianglar function and rational function. respectively, which include infinite sequences smooth solitary wave solutions, infinite sequences peak solitary wave solutions and infinite sequences compact soliton solutions. In this dissertation, beside K(m,n) equation, Degasperis-Procesi equation and CH equation, infinite sequences peak solitary wave solutions and infinite sequences compact soliton solutions are discovered in other nonlinear evolution equations.
Secondly. infinite sequences exact solutions of composite function.The infinite sequences exact solutions are composited by Jacobi elliptic function, hyperbolic function, trianglar func- tion and rational function in several forms, where include infinite sequences exact solutions are combined by smooth solitary wave solutions, peak solitary wave solutions and compact soliton solutions.
引文
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