Report Title: Introduction to Soliton Models in Mathematical Physics
Reporter: Qiao Zhijun
Affiliation: University of Texas
Report Time: December 15, 2025, 16:00-18:00
Report Location: Fifth Discussion Room, Mathematics Building, Jilin University
Report Abstract: Solitons are extremely peculiar phenomena in nature, exhibiting complex interactions governed by fundamental principles of mathematics and physics. Through this popular science report, one can gain an initial understanding in an accessible manner of the basic model of solitary water waves and their significant importance. The report first introduces the basic characteristics of solitary waves, emphasizing that they can maintain stable waveforms and constant propagation speeds even after interacting with other waves. It is precisely these unique properties that make solitary waves an extremely attractive subject of study in both theoretical and applied research.
We review the development of soliton models, starting from the Korteweg-de Vries (KdV) equation, and explain how it provides a fundamental understanding of one-dimensional soliton behavior in shallow water. The KdV equation elegantly captures the balance between nonlinearity and dispersion, which is the key factor in maintaining the isolated structure of waves. Additionally, we will introduce more complex models, such as the Burgers equation and the Camassa-Holm equation, which extend the understanding of solitons to more complex shallow water environments. These models help reveal the rich dynamical behaviors exhibited by wave propagation and interactions under different physical contexts. Finally, the report will briefly introduce our current interdisciplinary doctoral program in Mathematics and Statistics, as well as the teaching assistant (GTA) and research assistant (GRA) funding opportunities for international students.
Speaker Introduction: Dr. Qiao Zhijun is a Distinguished Professor at the University of Texas Rio Grande Valley (UTRGV). He earned his Master's degree from Zhengzhou University between 1986 and 1989, under the guidance of Professor Cao Cewen. In 1997, he obtained his Ph.D. from Fudan University, advised by Academicians Gu Chaohao and Hu Hesheng. His main research areas include nonlinear partial differential equations, integrable systems and nonlinear peakon waves, KdV equations and soliton theory, integrable symplectic mappings, R-matrix theory, image processing, and inverse problems in mathematical physics. In 1999, he was awarded the first National Hundred Outstanding Doctoral Dissertations. From 1999 to 2001, he served as a Humboldt Fellow at the University of Kassel in Germany. In 2013, he received the Distinguished Research Award from the University of Texas. In 2016, he was appointed as a Distinguished Professor at the University of Texas. In 2023, he was named an International Expert of the Fulbright Program. He has led over 20 national and international projects. He has published more than 150 academic papers in top international journals such as Communications in Mathematical Physics and IEEE Transactions on Geoscience and Remote Sensing (TGRS), and authored 2 monographs. Currently, he serves as a member of the editorial board of the international authoritative journal Studies in Applied Mathematics and as one of the editors-in-chief of the Journal of Nonlinear Mathematical Physics.
