报告题目:Nonlocal integrable systems and covariant hodograph transformations
报告人：张大军
报告时间：10：00-11：00
报告地点：理学楼教师活动中心
报告时间：6月2日10：00-11：00
理学楼教师活动中心
题目：Nonlocal integrable systems and covariant hodograph transformations
摘要： An integrable PT-symmetric system called nonlocal nonlinear Schrödinger (NLS) equation was proposed by Ablowitz and Musslimani [PRL-2013-No.064105]. It turns out that such nonlocal type integrable equations are derived from unreduced 2-component systems by using nonlocal reductions. In this talk we introduce a reduction approach to getting solutions of these nonlocal type integrable equations from the known solutions of those unreduced systems. Examples include semi-discrete NLS equation, and several nonlocal hierarchies with in the AKNS scheme. We also introduce other recent progress from other researchers such as formal transformation between local and nonlocal equations and reduction from matrix systems on half line.
We also introduce covariant hodograph transformations of nonlocal integrable systems. Short pulse (SP) equations serve as examples. First we describe connections between the first member in the AKNS negative hierarchy (AKNS(-1)) and several known integrable physical models, including Pedlosky's finite-amplitude baroclinic wave system, Konno-Oono system and its generalizations, SP equation and its complex and multi-component versions. These systems either are the AKNS(-1) system or can be derived from the system through suitable reductions and hodograph transformations. These connections can be extended to multi-component case. With this preparation we come to nonlocal reductions of the multi-component AKNS(-1) system and short pulse systems. In particular, we present nonlocal hodograph transformations between them.
Main References:
[1] K. Chen, D.J. Zhang, Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction, Appl. Math. Lett., 75 (2018) 82-8.
[2] X. Deng, S.Y. Lou, D.J. Zhang, Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schrödinger equations, arXiv:1707.07253.
[3] K. Chen, X. Deng, S.Y. Lou, D.J. Zhang Solutions of local and nonlocal equations reduced from the AKNS hierarchy, arXiv: 1710.10479. to appear in SAPM, DOI:10.1111/sapm.12215
[4] B. Yang, J.K. Yang, Transformations between nonlocal and local integrable equations, Stud. Appl. Math. 140:178-201 (2018).
[5] V. Caudrelier, Interplay between the Inverse Scattering Method and Fokas’s unified transform with an application, Stud. Appl. Math. 140:3-26 (2018).
报告人简介：张大军，上海大学数学系，教授、博士生导师。主要从事孤立子与可积系统等方面的研究。2004年起先后作为国家公派留学生和访问学者访问芬兰Turku大学物理系、英国Leeds大学等。先后指导10多位博士生，在国际期刊发表SCI论文90多篇，曾获上海市优秀博士论文，主持国家自然科学基金面上项目4项、教育部博士点基金1项。目前主要从事离散可积系统研究，担任国际系列会议SIDE (Symmetries and Integrability of Difference Equations)指导委员会委员。