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讲准字239号:Integrable systems with peakon and cuspon solution(具有尖峰解和尖点解的可积系统)

发布时间:2018-08-27|浏览次数:

题目:Integrable systems with peakon and cuspon solution(具有尖峰解和尖点解的可积系统)

主讲:乔志军

时间:2018年9月13日 10:00

地点:理学院201

主办:理学院


主讲简介:乔志军,美国德克萨斯大学大峡谷分校数学系教授。研究专长: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。美国德克萨斯大学数学学院首席教授,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年获全国百篇优秀博士论文,1999-2001年在德国卡塞尔大学数学系任洪堡学者。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。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;在全世界公认的五大数学物理学术组织担任会员;在数学物理领域内的六大一流国际期刊担任编委;将近百次被邀请在国际研讨会和专业学术会议上做主题发言与专题报告和学术报告,也在世界各地组织了20多次国际会议、研讨会。


主讲内容: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.


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