报告题目：Integrable peakon and cuspon equations

报告人： 乔志军教授   美国德克萨斯大学

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 hierarchy 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.

报告时间：2018年5月15日下午4:30-5:30

报告地点：数学科学学院三楼报告厅

乔志军教授简介： Zhijun (George) Qiao, 美国德州大学首席教授。1997年获得复旦大学博士学位,师从谷超豪院士和胡和生院士。研究方向是非线性偏微分方程,可积系统与非线性尖孤波,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。1999年获全国百篇优秀博士论文,1999-2001年获德国洪堡基金在德国Kassel大学任Humboldt Research Fellow。主持完成国家级项目20余项,在《Communications in Mathematical Physics》、《Journal of Nonlinear Science》、《Journal of Differential Equations》等著名国际刊物发表学术论文160余篇,出版著作2部,组织国际会议、研讨会20多次。