报告摘要

Abstract: 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 peakon equation is extended to the FORQ and other higher order peakon models with peakon and weak-kink solutions. In the end of my talk, some open problems are also addressed for discussion.

个人简介

乔志军教授于1997年获得复旦大学数学系博士学位，从师谷超豪院士和胡和生院士。1999年获得百篇优秀博士毕业论文。1997-2001任辽宁大学数学系教授。1999-2001，德国洪堡基金获得者。现任美国德克萨斯大学数学学院讲席教授。现有40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。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。现已出版著作2部，发表论文180余篇，其中包括著名国际杂志《数学物理学通讯》、《非线性科学》等。现作为项目负责人已经完成20多个国家项目。组织超过20个国际会议、研讨会。任国际期刊《Journal of Applied Analysis and Computation》和《Advances in Mathematical Physics》编委。