A seminar at Center for Mathematical Sciences,
June 18, Thursday, at 10:30am
报告人：乔志军教授，Department of Mathematics, University of Texas-Pan American
报告时间：2015年6月18日（星期四）上午10:30---11:30
报告地点：科技楼南楼702
报告题目：Integrable peakon and cuspon equations
报告摘要：In my talk, I will introduce integrable peakon and cuspon equations and present a basic approach how to get peakon solutions. Those equations include the well-known Camassa-Holm (CH), the Degasperis-Procesi (DP), and other new peakon equations with M/W-shape solutions. I take the CH case as a typical example to explain the details. My presentation is based on my previous work (Communications in Mathematical Physics 239, 309-341). I will show that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. In particular, the CH equation, constrained to a symplectic submanifold in $R^2N$, has the parametric solutions. Moreover, solving the parametric representation of the solution on the symplectic submanifold gives a class of a new algebro-geometric solution of the CH equation. In the end of my talk, some open problems are also addressed for discussion.
报告人简介：乔志军教授于1997年获得复旦大学数学系博士学位，师从谷超豪院士和胡和生院士，1999年获得首届全国百篇优秀博士毕业论文。1997-2001年任辽宁大学数学系教授。1999-2001年为德国卡塞尔综合大学数学系洪堡学者。现为美国德克萨斯大学数学系的终身教授。研究方向是非线性偏微分方程，可积系统与非线性水波，KdV方程和孤立子理论，可积辛映射，R-矩阵理论，雷达图像处理和数学物理的反问题。现已出版著作2部，发表论文130余篇，其中包括著名国际杂志Communications in Mathematical Physics, Journal of Nonlinear Science, SIAM Journal on Mathematical Analysis与Communications in Partial Differential Equations 等。目前主管研究生项目，现有30多位海外专家合作者, 已经指导5位博士后及超过20位研究生。作为项目负责人已经完成10多个国家项目。组织超过20个国际会议、研讨会。