KdV方程的精确多重波解研究

项目名称： KdV方程的精确多重波解研究

项目编号： No.11261001

项目类型： 地区科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 黄英

作者单位： 楚雄师范学院

项目金额： 28万元

中文摘要： 近年来，有不少研究者对KdV方程的孤立波解做了深入的研究，得到了大量的线状单孤子解和一些多重波解。但涉及多波解的大多数研究论文只是提供了由多个指数函数迭加或由Wronskian行列式表示的形式解，不是真正意义上的精确解，最终得到的精确多重波解并不多见。所以，由美国数学家M.J. Ablowitz和P.A. Clarkson利用逆散射法求出的几个线状多孤子解一直备受关注。而非线性的多波解几乎没有出现过，这是由于非线状多波解的复杂性和方法的适用性以及计算太繁琐等原因造成的。由于KdV方程的精确多重波解在分层内波、离子声波、等离子物理学和晶格力学等研究领域发挥着重要的作用。我们试图把常微分方程中常用的常数变易法创造性地运用于谷超豪先生关于达布变换的研究成果，进一步研究KdV方程的非线性精确多孤子解。并把研究孤子的方法推广性地运用于精确双周期解、三周期解、静态周期-孤子解和静态孤子-周期解的研究。

中文关键词： 贝克隆多变换；达布变换；多重波解；KdV 方程；AKNS 系统

英文摘要： The famous Korteweg de Vries (KdV) equation is a shallow water wave equation early derived by Korteweg de and Vries, its first application was discovered in the study of collision_free hydromagnetics waves in 1960. Following the further studies of new application, it has attracted much attention. Many researchers have carried out thorough investigations into the solitary wave solutions to the KdV equation in recent years, then a large number of linear one_soliton solutions and some many_wave solutions have been obtained. But most papers relative to many_wave solution only provided formal solutions which can be expressed in terms of either superposition of some exponential functions or Wronskian determinants, and these solutions are not true exact solution in its rigorous sense, thus the obtained exact solution of the KdV equation are not much. As a result, a few linear multi_soliton solutions including one_lump solution that were solved by American mathematicians M.J. Ablowitz and P.A. Clarkson with the inverse scattering transformation method have been well known for a long time. However, as we know, nonlinear exact many_wave solutions for the KdV equation have hardly appeared up to now, this result is caused by the complexity of nonlinear many_wave solution,the suitability of chosen method, tedious calculati

英文关键词： Backlund transformation；Darboux transformation；multiple wave solution；KdV equation；AKNS system