江苏省应用数学（中国矿业大学）中心系列学术报告
报告题目：On integrability of the q-deformed two-dimensional Toda lattice equation and its two-periodic reductions
报告人：李春霞，首都师范大学教授、博士生导师
报告时间：2023年4月28日9:00-10:00
腾讯会议：387-266-3917
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报告摘要：As is well-known, the two-dimensional Toda lattice equation (2DTLE) is extensively studied in literature. As an important extensionn of 2DTLE, it is of great interest to study the q-difference two-dimensional Toda lattice equation (q-2DTLE) which transforms to 2DTLE under the continuum limit q->1. In this talk, I will first propose a generalized Lax pair which involves several free parameters for 2DTLE. Next, starting from Darboux transformation for the noncommutative q-2DTLE, we construct the existing Casoratian solutions to the bilinear q-2DTLE and its bilinear Backlund transformation, which have already been obtained by Hirota’s method. And then, we successfully construct the binary Darboux transformation for the q-2DTLE, based on which, Grammian solutions expressed in terms of quantum integrals are established to the bilinear q-2DTL equation and its bilinear Backlund transformation simultaneously. This reveals the profound relation between Hirota’s method and Darboux transformation. In the end, as the 2-periodic reductions, a q-deformed sine-Gordon equation, a modified sine-Gordon equation together with their solutions are reported for the first time.
专家简介：李春霞，首都师范大学数学科学学院教授，博士生导师。2005年中科院数学院计算数学所博士，研究方向为孤子理论与可积系统。2005年-2007年清华大学博士后，2007-2008年受英国皇家学会资助在英国格拉斯哥大学从事博士后研究工作。先后作为访问学者访问英国剑桥大学牛顿数学科学研究所、美国University of South Florida和College of Charleston。主持国家自然科学基金和北京市自然科学基金5项。研究兴趣包括经典可积系统和非交换可积系统，可积系统与正交多项式、矩阵模型等不同数学分支的交叉。部分研究工作发表在Journal of Nonlinear Science, Proceedings of the Royal Society A, Journal of Physics A和Inverse Problems等上。