The Study Of Solutions To WBK And VCKdV Equations By Symbolic Computation

Soliton theory plays a very important role in various fields of natural science. On one hand, soliton theory has been widely used in the branches of quantum theory, particle physics, condensed matter physics, fluid physics, plasma physics and nonlinear optics, as well in the areas of mathematics, biology, chemistry, communications and other natural sciences; on the other hand, it greatly promotes the development of the theory of traditional mathematics, therefore, the investigation on soliton theory attracts great interest of physicists and mathematicians. With the in-depth of research and the development of science, especially the growing prosperity of nonlinear science, soliton theory has made a great progress, and more and more human and material resources have been focused on it. A large number of papers and magazines in this area have appeared, as well as many international academic conferences on this point have been held one after another. For example, the conferences of "The nonlinear soliton structure and dynamics in the condensed matter physics" has been held in Oxford, and "Solitons in physics" in Goteborg. Studies on soliton theory in China has been started in the 1970's. At that moment, Profs. Zhen Ning Yang, Zhen Dao Lee, Xing Shen Chen and other domestic counterparts introduced the development progress of soliton theory abroad and pointed out its importance. Afterwards, Chinese Academy of Sciences and several higher education schools gradually carried out research works in this area. Small seminars respectively held in 1980 (Xiamen) and 1986 (Shanghai) have promoted the research activities of the soliton theory.Based on the theory of nonlinear partial differential equations and with the help of symbolic computation, this paper makes some research on the solution methods of the Whitham-Broer-Kaup (WBK), variable-coefficient Korteweg-de Vries (KdV), variable coefficient nonlinear Schrodinger equations.The chapters and contents are as follows:The first chapter introduces the history of solitons, progrees of soliton theory and research methods frequently used in nonlinear physical equations.The second chapter focuses on the knowledge of Darboux transformation, Lax Pair and Ablowitz-Kaup-Newell-Segur (AKNS) system, which are studied during my postgraduate terms.The third chapter investigates the WBK equation and its physical background, then uses Gauge transformation method to get the relations between the WBK equation and the AKNS system, as well between their solutions.The fourth chapter introduces four basic transformation which are used to derive the Lax pair and BT for the variable-coefficient KdV and Schrodinger equations.