【作者】 包霞；

【导师】 斯仁道尔吉；

【作者基本信息】 内蒙古师范大学 ， 科学技术史， 2019， 博士

【摘要】 1834年8月,英国爱丁堡大学的数学教授、优秀的造船工程师罗素在校园附近的联合运河中首次观察到孤立波。1965年,美国数学家克鲁斯卡尔和扎布斯基通过计算机模拟了孤立波的“碰撞”,发现经碰撞后的它们不会改变形状、大小和方向。于是,二人在《Physical Review Letters(物理评论快报)》上发文首次提出了“Soliton”(孤立子)这个名词,以此来强调孤立波的“粒子”性行为与特性,标志着孤立子理论的正式诞生。随着计算机技术的不断发展,人们在物理学、生物学、医学、海洋学、经济学、人口问题等诸多领域都发现了孤立子及与其密切相关的重要问题,孤立子成为非线性科学的三大普适类之一。20世纪70年代后,孤立子理论传入国内,学者们在高校科研院所里开始进行孤立子的研究,先学习国外已有理论成果,再进行有效拓展和理论创新,同时注重培养自己的研究生。这是一个积极良性互动的学习过程,在短短十年里就取得了可喜的成绩,也进一步促进了理论的传播与发展。孤立子理论在中国的研究与发展虽然之前也受到近现代数学史研究者的关注,但是在谈及20世纪数学科学的回顾时基本没有提到孤立子理论的研究与发展,更没有从数学史的角度进行系统的梳理研究,这就无法全面地反映出中国现代数学的研究全貌。因此,本文“孤立子理论在中国的发展(1978-1989)”便具有重要的理论和现实意义。在查阅了大量原始资料和现有研究文献,并采访一些老一辈学者,采用文献分析、归纳分析、调研实践等方法,对中国孤立子理论研究做了较系统的分析总结:1.结合孤立子理论的四个发展阶段,论述1834至1989年间世界孤立子理论研究的主要成果及其意义。2.考查了中国学者在国内外发表的孤立子理论研究论文和已有的研究文献,经过细致筛选,介绍了谷超豪、屠规彰、李翊神、曹策问、郭柏灵等代表性学者的求学之路及学术研究概况,同时介绍了学界其他学者的一些重要研究成果。通过分析归纳,本文首次较为全面地阐述了屠规彰等人的孤立子理论研究工作;总结了中国在孤立子理论领域的主要研究成果,包括反散射方法、B?cklund变换法、Darboux变换法、守恒律、对称及其代数结构、Lax对的非线性化、屠格式、孤子方程的规范等价分类、孤立子的实验数值研究等领域;分析了中国孤立子理论研究的特征及其贡献。3.统计了二十世纪七八十年代在国际上具有影响力的孤立子研究著作。基于中国第一部孤立子理论译著和第一部理论专著的重要性,对这两本书进行了介绍,发掘其历史价值与学术意义。4.通过对前辈的访谈和研读他们留下的手稿和研究文献,尝试梳理出中国孤立子理论研究学者开展的活动,包括全国孤立子与可积系统研讨会、国内主要科研院所的教研、参加国际学术会议,与国外学者的学术交流,从中分析这些活动对中国孤立子理论研究的影响。5.在翻阅大量文献资料的过程中,得到借鉴与启发,进一步探究孤立子理论,构造了KP型方程的新型Darboux变换和广义变系数KdV方程的Lax方程组的求解递推公式,在实践意义上实现了研究数学史的目的之一。本论文包括六章内容。第一章:孤立子理论的发展概况(至1989年)。这一章根据孤立子理论发展的四个阶段,较详细地论述了从孤立波被发现到1989年第三阶段结束的主要研究成果。第一阶段(1834-1954)包括孤立波的发现(1834)、孤立波的数学模型——KdV方程的提出(1895)、Boussinesq方程的提出(1872)、sine-Gordon方程的B?cklund变换(1885)、Cole-Hopf变换(1950,1951)等;第二阶段(1955-1970)包括FPU实验(1955)、孤立子的发现(1965)、怪波理论(1965)、反散射方法的提出(1967)、Lax对特征值问题(1968)、KP方程的提出(1970)等;第三阶段(1971-1989)包括Hirota双线性方法(1971)、光孤子的发现(1973)、延拓结构法(1975)、偏微分方程的Painlevé分析方法(1983)、Lax对的非线性化(1989)、屠格式(1989)等。第二章:孤立子理论在中国的发展概况(1978-1989)。这一章首先从国内外环境阐述了孤立子理论传入中国的起始,考查了国内第一篇关于孤立子理论研究论文的内容和意义,其次再现并阐述了中国孤立子理论研究的代表性学者屠规彰、李翊神、曹策问、郭柏灵、谷超豪等人的求学之路及学术研究概况,最后统计了在世界上具有影响力的孤立子理论著作及中国学者的译著与专著。第三章:中国孤立子理论研究学者开展的活动。本章首先介绍了国内孤立子理论主要研究团队的教研情况,并对中国第一部孤立子理论译著与第一部理论专著分别进行介绍。然后转向与国外学界的互动交流方面,介绍了去海外参加国际学术会议和访学的中国孤立子理论研究学者。第四、五章是中国孤立子理论研究学者开展的具体研究内容——非线性演化方程的孤立子解的求法和解的适定性研究及可积系统研究。首先重点讲述了国内主要研究的非线性演化方程的四种解法:B?cklund变换法(BT)、Darboux变换法(DT)、反散射方法(IST)、Hirota方法的研究背景和国内外发展概况及中国学者的主要研究成果。另外,在梳理中国孤立子理论的过程中也不断受到启发,就其中的Darboux变换法的理论研究进行了新的拓展。其次,从孤子方程的可积性判别、孤子方程的规范等价类、构造有限维可积系统的有效方法—Lax对的非线性化方法、构造无限维可积系统的有效方法——屠格式、寻找守恒律及守恒律个数的猜想证明、构造对称及其代数结构研究等六个方面,详细介绍了国内学者的探讨过程和研究成果。第六章:孤立子的实验数值研究。本章阐述了国内学者在孤立子的实验数值研究方面的突出工作:首先是,吴君汝通过实验发现了非传播的孤立波,该波后来被命名为“吴氏波”(或吴立子)。吴氏孤波的发现证实了孤立波也可能是非传播性的波,而非传播的孤立波比传播的孤立波更具稳定性和重复性,所以它的发现被认为是当代非线性波研究的重大进展。其次是郭本瑜在孤立子解的数值计算方面的工作及成果介绍。总之,本文通过文献考证和文献分析方法,考察分析了国内早期(1978-1989)孤立子理论的论著、名人传记及研究性论文,综述孤立子理论在中国的早期传播、研究与发展,认为1978—1989年这一时期我国孤立子理论研究主要处于培养人才和学习阶段,是迎接孤立子理论在中国大发展的筹备期。在此阶段出现了屠规彰的“屠格式”、曹策问的“Lax对的非线性化方法”、谷超豪的“Darboux矩阵法”等可纳入国际孤立子理论研究前沿的可喜成果且这些方法至今仍广泛应用于可积系统的构造和非线性演化方程求解,是非常有效的方法。更多还原

【Abstract】 In August 1834,Russell,a distinguished naval architect and mathematics professor at the University of Edinburgh,first observed isolated waves in the Union Canal near the campus.In 1965,American mathematicians M.D.Kruskal and N.J.Zabusky simulated the“collision”of solitary waves by computer,and found that the waves did not change their shape,size and direction after the collision.Therefore,they proposed“Soliton”for the first time in Physical Review Letters to emphasize its particle property,marking the official birth of the soliton theory.With the development of computer technology,soliton and its related important problems have been found in many fields such as physics,biology,medicine,oceanography,economics and population.Soliton has become one of the three general categories of nonlinear science.In the 1970s,with the introduction of the soliton theory into China,scholars started research on solitons at the research institutes of universities and colleges by studying the theoretical achievements abroad,expanding and innovating theoretically and cultivating postgraduate students.It is proved to be a positive and interactive learning process,with gratifying achievements made and the theory further developed and spread in just ten years.Although the study and development of the soliton theory in China has aroused some concern of the modern researchers of the history of mathematics,it is not mentioned in the overviews of the mathematical science of the 20~thh century,let alone is studied systematically in the perspective of mathematics history.Therefore,the existing literature cannot reflect the domestic research status fully.Thus,the soliton theory in China(1978-1989)has an important significance in theories and practices.The author has consulted a large number of original materials and existing research literatures,and interviewed some highly respectable senior scholars at the same time.By means of literature analysis,inductive analysis and research practice,the author has also made a systematic summary of the research on Chinese soliton theory.The specific work is as follows:1.This paper divides the development of the soliton theory into four stages,and discusses the main achievements and meaning of the global soliton theory from 1834 to 1989.2.The author reviewed the related papers and literatures of Chinese scholars published at home and abroad.After careful screening,the author introduces the education experience and relevant research works of some representative scholars like Tu Guizhang,Li Yishen,Cao Cewen,Guo Boling,Gu Chao-hao,and the important research results of some other scholars.Through analysis and induction,this paper expounds the research work of Tu Guizhang et al.for the first time.The main research results of China are summarized,including the inverse scattering method,Hirota bilinear method,B?cklundtransformationmethod,Darbouxtransformationmethod,conservation law,symmetry and its algebraic structure,nonlinearization of Lax pair,Tu scheme,gauge equivalence classification of soliton equations,experimental and numerical study of solitons,etc.And then,the characteristics and contributions of Chinese researchers on soliton theory are introduced.The paper has counted the internationally influential works on solitons from 1970s to 1980s.Based on the importance of the first translation of soliton theory and the first theoretical monograph in China,the author introduced the historical value and academic significance of two books of solitons.After interviewing the predecessors and studying their manuscripts,the author tried to tease out the activities participated by Chinese scholars,including the national seminar on soliton,the research of main research institutes and academic exchanges and analyzed their influence on the research of Chinese soliton theory.By reviewing a large number of literature materials,the author further explored the soliton theory and gained insight into Darboux transformation,thus fulfilling one of the purposes of studying the history of mathematics in the practical context.This thesis includes six chapters.Chapter one is an overview of the development of the soliton theory(up to 1989).This chapter,according to the four stages of the development of soliton theory,expounds the main research results from the discovery of soliton waves to the end of the third stage in 1989.The first stage(1834-1954)embraces the discovery of solitary waves(1834),the mathematical model of solitary waves--the derivation of KdV equation(1895),the proposal of Boussinesq equation(1871),the B?cklund transformation of sine-Gordon equation(1885),Cole-Hopf transformation(1950,1951),etc.The second stage(1955-1970)includes FPU experiment(1955),the discovery of solitons(1965),rogue wave theory(1965),the proposal of inverse scattering method(1967),eigenvalue problems of Lax pair(1968),KP equation(1970),etc.The third stage(1971-1989)contains Hirota bilinear method(1971),discovery of optical solitons(1973),the method of prolongation structure(1975),the Painleve analysis of partial differential equations(1983),the nonlinearization of Lax pair(1989),Tu scheme(1989),etc.Chapter two explores the development of soliton theory in China(1978-1989).First,this chapter expounds the initiation of soliton theory in China,and investigates the first domestic paper on the soliton theory.Then,it enumerates some representative scholars in the study of Chinese soliton theory like Tu Guizhang,Li yishen,Cao cewen,Guo boling,Gu chaohao,and introduces their course of studying and academic research.Finally,it covers statistics on some influential soliton theory books around the world and translations and monographs of Chinese scholars.Chapter three probes into the activities done by Chinese scholars of solition theory.This chapter firstly introduces the teaching and research situation of the domestic main research teams,and then comments on the first Chinese translation of soliton theory and the first Chinese theoretical monograph in this field.Turning to the interaction and communication aspect,the author introduces some Chinese scholars of soliton theory who have been abroad to participate in international academic conferences or been visiting scholars to overseas universities.Chapter four and five are the specific contents of research carried out by Chinese scholars of soliton theory--the research on the integrable systems and the soliton solutions of the nonlinear evolution equations.Firstly,it focuses on four research backgrounds to the nonlinear evolution equation mainly studied in China:B?cklund transformation method(BT),Darboux transformation method,inverse scattering method(IST),the Hirota method.The main research results of Chinese scholars are also described.In addition,in the process of sorting out the Chinese soliton theory,the author was also constantly inspired,and did some further research on Darboux transformation.Then,the paper introduces the research process and results of the domestic scholars from these aspects:distinction of integrability of soliton equation,the gauge equivalence relation of soliton equation,the nonlinearization of Lax pair for constructing finite dimensional integrable systems,the Tu scheme for constructing infinite dimensional integrable systems,the method for seeking conservation law and the proof of the number of conservation laws and the studies on the symmetric structures and their algebraic structures.Chapter six is on the experimental numerical study of solitons.This chapter describes the outstanding work of domestic scholars in the experimental numerical study of solitons.First,Wu Junru discovered the non-propagating solitons through experiments,which was later named“Wu’s wave”(or“Wu soliton”).The discovery of Wu’s solitary waves has confirmed that solitary waves may also be non-propagating waves,and non-propagating solitary waves are more stable and repeatable than propagating solitary waves,so its discovery is considered as a significant progress in the study of contemporary nonlinear waves.Then,Guo benyu’s work and achievements in the numerical calculation of soliton solutions are introduced.In the way of textual research,this article analyzes the domestic works,celebrity biographies and research papers in the early development of the soliton theory,then summarizes the early spread,research and development of soliton in China.It also points out that the period from 1978 to 1989 is just the period for the field of soliton in China to cultivate talents and study,the early stage for the oncoming academic achievement,and the preparatory period for the soliton theory to move toward the forefront of the world.更多还原