本科毕业论文带有外加势的变系数扩展KdV方程的可积性研究和孤子解.doc

带有外加势的变系数扩展 KdV 方程的可积性 研究和孤子解 李敏1，肖井华1，王明1，王玉凤1，田播1，2 1 北京邮电大学理学院，北京　100876 2 北京邮电大学信息光子与光通信国家重点实验室，北京 100876 摘要：本文研究了带有外加势的变系数扩展 KdV 方程，它可以用来描述流过障碍物时的分层 流体跨临界流。通过计算机符号计算，构造了方程的非等谱的 AKNS 系统。并给出了广义的 可积条件，在这个条件下，该方程可以退化为可积的变系数 KdV 方程和扩展 KdV 方程。借 助双贝尔多项式方法，在可积条件下获得了方程的单孤子和双孤子解。分析了变系数效应对扭 结型和钟形孤子的影响。最后，在不同的变系数条件下给出了几种类型的孤子相互作用情况。 关键词：偏微分方程；可积性；孤子解；符号计算；双贝尔多项式 中图分类号： O175; O29 Integrability and soliton solutions of a forced extended Korteweg-de Vries equation with variable coe?cients Li Min1, Xiao Jing-Hua1, Wang Ming1, Wang Yu-Feng1 and Tian Bo1,2 1 2 School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876 State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing, 100876 Abstract: Under investigation is a forced extended Korteweg-de Vries (KdV) equation with time- and space-dependent variable coe?cients, which can describe the transcritical ?ow of a strati?ed ?uid over an obstacle. Nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) system for this equation is constructed via the symbolic computation. General integrable conditions are given, under which this equation can be reduced to such integrable equations as the KdV and extended KdV equations with variable coe?cients. One- and two- soliton solutions are 基金项目： National Natural Science Foundation of China ，Fundamental Research Funds for the Central Universities of China (2011BUPTYB02)，National Basic Research Program of China (973 Program) (2010CB923200)，Spe- cialized Research Fund for the Doctoral Program of Higher Education (200800130006)，and by the Beijing University of Posts and Telecommunications Excellent Ph.D. Students Foundation (CX201111). 作者简介： Li Min（1985-），female，postgraduate student，major research direction：nonlinear science; Xiao Jing-Hua （1965-），male，professor，major research direction：nonlinear science; Wang Ming（1988-），female，postgraduate student， major research direction：nonlinear