报告1：
报告题目：Progresses on Some Open Problems Related to Infinitely Many Symmetries
报告人：楼森岳 教授（宁波大学）
报告时间：2024年11月15日（周五）下午14：00
主持人：刘文军 教授
报告地点：北邮沙河校区理学楼520室
报告摘要：
The quest to reveal the physical essence of the infinitely many symmetries and/or conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of n-wave solutions, such as the n-soliton solutions, multiple breathers, complexitons, and the n-periodic wave solutions (the algebro-geometric solutions with genus n), wherein n denotes an arbitrary integer that can potentially approach infinity. Each subwave comprising the n-wave solution may possess free parameters, including center parameters ci, width parameters (wave number) ki, and periodic parameters (the Riemann parameters) mi. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinitely many symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. This conjecture appears to hold true for all integrable systems with n-wave solutions. The conjecture intimates that the currently known infinitely many symmetries is not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned Korteweg–de Vries (KdV) equation and the Burgers equation as simple examples, the conjecture is substantiated for the n-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative, which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework. Notably, a ren-symmetric integrable Burgers hierarchy is explicitly derived. Both the supersymmetric and the classical integrable hierarchies are encompassed within the ren-symmetric integrable hierarchy. The results of this paper will make further progresses in nonlinear science: to find more infinitely many symmetries, to establish novel methods to solve nonlinear systems via symmetries, to find more novel exact solutions and new physics, and to open novel integrable theories such as the ren-symmetric integrable systems and the possible relations to fractional integrable systems.
报告人简介：
楼森岳，教授，宁波大学博士生导师，973项目科学家，国家“有突出贡献中青年科技专家”，国家“百千万人才工程一、二层次人选”，国家杰出青年基金获得者，享受国务院政府特殊津贴，新世纪“151”人才工程第一层次人选。曾任《Communication in Theoretical Physics》杂志和《Chinese Physics Letters》杂志的编委，上海交通大学物理系兼职博士生导师。曾获国家教委科技进步二、三等奖和上海市科技进步二等奖。
主要研究领域：量子场论、粒子物理和非线性物理。特别在非线性物理可积体系的研究中作出了一些具有独创性的工作。如：建立了求解非线性方程的形变映射方法；建立了1+1维可积体系强对称算子的因式化和逆方法；建立了形式级数对称理论；建立了无穷多Lax对和非局域对称理论；给出了多种意义下的高维可积模型；在实验上观察到了宏观格点体系的多种孤子激发模式；建立了多线性分离变量法和导数泛函分离变量法等等。主持国家基金重点项目2项。
报告2 ：
报告题目：双线性方法与可积性
报告人：张大军 教授（上海大学）
报告时间：2024年11月15日（周五）下午14：00
主持人：刘文军 教授
报告地点：北邮沙河校区理学楼520室
报告摘要：
1971年,Ryogo Hirota 首创双线性方法，获得了KdV方程的多孤子解相比于GGKM的反散射变换，Hirota的双线性方法可以称为直接方法。它不仅在可积系统的精确求解中显示出强大的功能，而且由Hirota引入的双线性导数（算子）以及可积系统的双线性形式，在可积系统理论的研究中扮演着独特的角色。由双线性方法引出的上世纪80年代由日本京都数学所M.Sato等学者发展起来的著名的Sato理论，揭示了可积系统及其双线性形式深刻的数学结构；作为双线性方程解的tau函数不断出现于数学物理的众多分支中。此报告旨在对双线性方法给一个非常初步的介绍，内容侧重于双线性方法与可积性的联系，主要涉及：
1. 2-孤子解的普遍存在性；
2. 3-孤子解与Hirota可积性；
3. Bäcklund变换与非线性叠加公式；
4. tau函数的顶点算子表示。
报告人简介：
张大军教授，上海大学数学系教授，博士生导师。主要从事离散可积系统与数学物理的研究，在离散可积系统的直接方法、多维相容性的应用、空间离散下的可积结构与连续对应、精确解的结构与应用等方面取得了有意义的学术成果。曾作为访问学者，访问Turku大学、Leeds大学、剑桥牛顿数学研究所、Sydney大学、早稻田大学等学术机构。先后主持国家自然科学基金面上项目和国际合作项目7项、参与国家自然科学基金重点项目1项。曾担任国际期刊Journal of Nonlinear Mathematical Physics编委。目前担任离散可积系统国际系列会议SIDE (Symmetries and Integrability of Difference Equations)指导委员会委员和国际期刊Journal of Physics A和Open Communications in Nonlinear Mathematical Physics编委。
