The Development Of Soliton Theory In China (1978-1989)

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.