Speaker: 袁小平 (Fudan University)
Time: 2025-10-16 16:30-17:30
Location: 1310, SIMIS
Abstract: Dealing with unbounded perturbations of integrable partial differential equations (PDEs), KAM theory encounters significant difficulties. The first breakthroughs in handling such unbounded perturbations were pioneered by Kuksin who established the Kuksin Lemma and developed applicable KAM theorems to analyze the persistence of finite-gap solutions for the KdV equation. Subsequently, the Kuksin Lemma was extended to its limiting case and KAM theorems were constructed for the derivative nonlinear Schrödinger equations (DNLS), and Benjamin-Ono equations. At almost the same time, KAM theorems were constructed by the Italian school for some quasi-linear PDEs of spatial dimension 1. In this talk, we address some recent progresses about KAM for some quasi-linear PDEs of multi-dimension
About Speaker:
袁小平是复旦大学数学科学学院特聘教授、博士生导师，长期从事常微分方程与动力系统研究，尤其在KAM理论及无穷维哈密顿系统领域取得突出成果。他是国家杰出青年科学基金获得者，教育部长江学者特聘教授，享受国务院政府特殊津贴。他发展了无穷维KAM理论体系，并成功应用于构造非线性发展方程（如KdV方程、Schrödinger方程）的不变环面与拟周期解，提出超指数衰减的呼吸子，该理论预测后被物理实验证实，并与学生合作证明了量子Duffing振子的纯点谱性质。
Yuan Xiaoping is a Distinguished Professor and a doctoral advisor at the School of Mathematical Sciences, Fudan University. He has long been engaged in research on ordinary differential equations and dynamical systems, and has achieved outstanding results, particularly in the fields of KAM theory and infinite-dimensional Hamiltonian systems. He is a recipient of the National Science Fund for Distinguished Young Scholars, a Distinguished Professor under the Chang Jiang Scholars Program of the Ministry of Education, and entitled to the Special Government Allowance of the State Council. He developed a theoretical framework for infinite-dimensional KAM theory and successfully applied it to the construction of invariant tori and quasi-periodic solutions for nonlinear evolution equations (such as the KdV equation and the Schrödinger equation). He proposed the Breather with super-exponentially decay, a prediction of which was later confirmed by physical experiments. Additionally, he collaborated with his students to prove the purely point spectrum property of the quantum Duffing oscillator.
简体中文 
English