報告1：
報告題目：Progresses on Some Open Problems Related to Infinitely Many Symmetries
報告人：樓森岳 教授（寧波大學(xué)）
報告時間：2024年11月15日（周五）下午14：00
主持人：劉文軍 教授
報告地點：北郵沙河校區(qū)理學(xué)樓520室
報告摘要：
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.
報告人簡介：
樓森岳，教授，寧波大學(xué)博士生導(dǎo)師，973項目科學(xué)家，國家“有突出貢獻中青年科技專家”，國家“百千萬人才工程一、二層次人選”，國家杰出青年基金獲得者，享受國務(wù)院政府特殊津貼，新世紀“151”人才工程第一層次人選。曾任《Communication in Theoretical Physics》雜志和《Chinese Physics Letters》雜志的編委，上海交通大學(xué)物理系兼職博士生導(dǎo)師。曾獲國家教委科技進步二、三等獎和上海市科技進步二等獎。
主要研究領(lǐng)域：量子場論、粒子物理和非線性物理。特別在非線性物理可積體系的研究中作出了一些具有獨創(chuàng)性的工作。如：建立了求解非線性方程的形變映射方法；建立了1+1維可積體系強對稱算子的因式化和逆方法；建立了形式級數(shù)對稱理論；建立了無窮多Lax對和非局域?qū)ΨQ理論；給出了多種意義下的高維可積模型；在實驗上觀察到了宏觀格點體系的多種孤子激發(fā)模式；建立了多線性分離變量法和導(dǎo)數(shù)泛函分離變量法等等。主持國家基金重點項目2項。
報告2 ：
報告題目：雙線性方法與可積性
報告人：張大軍 教授（上海大學(xué)）
報告時間：2024年11月15日（周五）下午14：00
主持人：劉文軍 教授
報告地點：北郵沙河校區(qū)理學(xué)樓520室
報告摘要：
1971年,Ryogo Hirota 首創(chuàng)雙線性方法，獲得了KdV方程的多孤子解相比于GGKM的反散射變換，Hirota的雙線性方法可以稱為直接方法。它不僅在可積系統(tǒng)的精確求解中顯示出強大的功能，而且由Hirota引入的雙線性導(dǎo)數(shù)（算子）以及可積系統(tǒng)的雙線性形式，在可積系統(tǒng)理論的研究中扮演著獨特的角色。由雙線性方法引出的上世紀80年代由日本京都數(shù)學(xué)所M.Sato等學(xué)者發(fā)展起來的著名的Sato理論，揭示了可積系統(tǒng)及其雙線性形式深刻的數(shù)學(xué)結(jié)構(gòu)；作為雙線性方程解的tau函數(shù)不斷出現(xiàn)于數(shù)學(xué)物理的眾多分支中。此報告旨在對雙線性方法給一個非常初步的介紹，內(nèi)容側(cè)重于雙線性方法與可積性的聯(lián)系，主要涉及：
1. 2-孤子解的普遍存在性；
2. 3-孤子解與Hirota可積性；
3. B?cklund變換與非線性疊加公式；
4. tau函數(shù)的頂點算子表示。
報告人簡介：
張大軍教授，上海大學(xué)數(shù)學(xué)系教授，博士生導(dǎo)師。主要從事離散可積系統(tǒng)與數(shù)學(xué)物理的研究，在離散可積系統(tǒng)的直接方法、多維相容性的應(yīng)用、空間離散下的可積結(jié)構(gòu)與連續(xù)對應(yīng)、精確解的結(jié)構(gòu)與應(yīng)用等方面取得了有意義的學(xué)術(shù)成果。曾作為訪問學(xué)者，訪問Turku大學(xué)、Leeds大學(xué)、劍橋牛頓數(shù)學(xué)研究所、Sydney大學(xué)、早稻田大學(xué)等學(xué)術(shù)機構(gòu)。先后主持國家自然科學(xué)基金面上項目和國際合作項目7項、參與國家自然科學(xué)基金重點項目1項。曾擔任國際期刊Journal of Nonlinear Mathematical Physics編委。目前擔任離散可積系統(tǒng)國際系列會議SIDE (Symmetries and Integrability of Difference Equations)指導(dǎo)委員會委員和國際期刊Journal of Physics A和Open Communications in Nonlinear Mathematical Physics編委。
