时间：2022年10月14-15日

地点：腾讯号：268-691-439   会议密码: 202210

主办单位：中国科学院青年创新促进会数理分会

　　　　　中国科学院数学与系统科学研究院

　　　　　北京大学

 

 

报告人: 程纪鹏（中国矿业大学）

报告题目: 自由费米子与可积系统

摘要: 本报告将主要介绍KP、modified KP与Toda方程族的自由费米子构造，并讨论自由费米子在附加对称与达布变换中的一些应用。

 

报告人: 陈金兵（东南大学）

报告题目: Periodic standing waves in the DNLS equation: modulational instability and rogue waves

摘要: The DNLS equation is the canonical model for dynamics of nonlinear waves in plasma physics and optics. We study exact solutions describing rogue waves on the background of periodic standing waves in the DNLS equation. We show that the space-time localization of a rogue wave is only possible if the periodic standing wave is modulationally unstable. If the periodic standing wave is modulationally stable, the rogue wave solutions degenerate into algebraic solitons propagating along the background and interacting with the periodic standing waves.

 

报告人：黄晴（西北大学）

报告题目: Integrable Extensions of the Rational Calogero-Moser Model and Related Stationary Flows

摘要: We consider a class of Hamiltonian systems in 3 degrees of freedom, with a particular type of quadratic integral and which includes the rational Calogero-Moser system. We introduce separation coordinates to find the general Liouville integrable system. This gives a coupling of the Calogero-Moser system with a large class of potentials, generalising the series of potentials which are separable in parabolic coordinates. Particular cases are superintegrable, including Kepler and a resonant oscillator. Meanwhile, we study the Lax representations of the generalised systems by relating them to the stationary flows of integrable PDEs.
   This is a joint work with Allan P. Fordy.

 

报告人: 郎红蕾（中国农业大学）

报告题目: Rota-Baxter Lie groups

摘要: Rota-Baxter operators on Lie algebras were first studied by Belavin, Drinfeld and Semenov-Tian-Shansky as operator forms of the classical Yang-Baxter equation. As a fundamental tool in studying integrable systems, the factorization theorem of Lie groups by Semenov-Tian-Shansky was obtained by integrating a factorization of Lie algebras from solutions of the modified Yang-Baxter equation. Integrating the Rota-Baxter operators on Lie algebras, we introduce the notion of Rota-Baxter operators on Lie groups and more generally on groups. Then the factorization theorem can be achieved directly on groups. This is joint work with Li Guo and Yunhe Sheng.

 

报告人：李传忠（山东科技大学）

报告题目: Reductions and generalizations of Toda systems

摘要: In this talk, we will review a series of our studies on all kinds of reductions, and generalizations on the Toda systems, including triangular Toda hierarchy,  two-dimensional Toda hierarchy, bigraded Toda hierarchy(BTH), B(C) type Toda  hierarchy, supersymmetric Toda lattice hierarchy and finite-sized matrix-valued Toda systems. These studies contain their symmetries, Darboux transformations, tau functions and so on. We give the rational solutions of finite-sized matrix-formed Toda system which are expressed by the products of the Schur polynomials corresponding to non-rectangular Young diagrams.  Interestingly, we consider a regular solution for the (1,2)-BTH with 3-sized Lax matrix, and discuss some geometric structure of the solution from which the difference between (1,2)-BTH and original Toda hierarchy is shown.  In the case of the original Toda lattice, one can show that all the diagonal orbits have to cross the center point. However, the orbits for the (1,2)-BTH have no such restriction, but those still go through the points close to the center.

 

报告人：李世豪（四川大学）

报告题目: On Pfaff lattice

摘要: 本报告将从可积系统与随机矩阵的联系出发，介绍什么是Pfaff格，以及Pfaff格的代数特点。更多地，我将从斜正交多项式的角度讲述Pfaff格的可积结构，然后介绍自己在Pfaff格上的一些相关工作。

 

报告人: 凌黎明（华南理工大学）

报告题目: Darboux transformation and soliton solution to the coupled complex short pulse equation

摘要: The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright-soliton, double-hump vector solitons to the CCSP equation. Meanwhile, we perform the inverse scattering analysis to the vanishing background.  (joint with Prof. Baofeng Feng）

 

报告人：刘家琪（中国科学院大学）

报告题目: 低正则性空间与黎曼希尔伯特方法——以mkdV和sine-gordon方程为例

摘要：mkdv 和sine-gordon都是可积偏微分方程的著名代表，本人将讨论近期与合作者取得的一系列低正则性和稳定性的结果，重点在于如何结合反散射方法和pde方法以求取pde视角的完整结果。合作者为陈功和陆冰滢。

 

报告人：田凯（中国矿业大学（北京））

报告题目: A supersymmetric generalized NLS equation: prolongation algebra, linear spectral problems and beyond

摘要: For a supersymmetric generalized nonlinear Schr？dinger (NLS) equation, the Wahlquist-Estabrook prolongation algebra L is calculated, and successfully embedded in the Lie superalgebra A(1,1). A non-trivial matrix realization of L is established. Consequently, a proper linear spectral problem is constructed for the supersymmetric NLS equation. We also derive its integrable properties, including a recursion operator, Hamiltonian structure. This is a joint work with Prof. Q. P. Liu and Dr. Liming Zang.

 

报告人：王东（中国科学院大学）

报告题目： Muttalib-Borodin系综的Riemann-Hilbert问题求解中的一个local parametrix构造

摘要：在和张仑一起研究Muttalib-Borodin系综的过程中，我们需要构造一个新的local parametrix。这个构造过程也许有独立的意义。本报告将解释这个local parametrix的缘起和具体形式。同时也介绍Muttalib-Borodin系综，一种与随机矩阵有联系的，来自quantum transport的模型。

 

报告人：王灯山（北京师范大学）

报告题目: Rarefaction problem of the focusing nonlinear Schr？dinger equation

摘要: In this talk, the long-time asymptotic behaviors of two separable plane waves of the focusing nonlinear Schr？dinger equation in the presence of discrete spectrum are analyzed via the Riemann-Hilbert formulation. It is found that for large time there are three kinds of asymptotic regions in space-time frame, which includes the plane-wave region with soliton transmission from the discrete spectrum, the elliptic wave region with a one-phase wave and the cavitation region. The leading-order terms for these regions are computed with error estimates by using the Deift-Zhou steepest-descent method for Riemann-Hilbert problems. The numerical simulations match well with the theoretical analysis.

 

报告人：吴朝中（中山大学）

报告题目: Drinfeld-Sokolov 方程簇的拓扑解

摘要: 对任意非扭仿射 Kac-Moody 代数，我们通过求解相应的约化弦方程来计算 Drinfeld-Sokolov 方程簇的拓扑解。特别地，仿射 $A_1^{(1)}$ 情形的拓扑解对应著名的 Witten-Kontsevich tau 函数。

 

报告人：吴志伟（中山大学）

报告题目: Geometric curve flows related to integrable systems

摘要: In this talk, I will give a brief introduction on constructing curve flows whose invariants are related to certain soliton hierarchies and how to study its integrable properties, such as bi-Hamiltonian formulation and Darboux transformation. As an example, we will discuss the case of Lagrangian curve flows. This talk is based on a series of joint works with Chuu-Lian Terng.

 

报告人: 杨迪（中国科学技术大学）

报告题目: 可积系统中的tau-函数

摘要: In this talk, we discuss the notion of tau-functions for integrable systems. We will present the ideas and our approach mainly for the KdV hierarchy. More precisely, for an arbitrary solution to the KdV hierarchy we derive an explicit formula of the generating series of logarithmic derivatives of the tau-function in terms of matrix resolvents. Then, using the wave functions, we derive two additional explicit formulae. Applications to several classes of enumerative invariants are given. The talk is based on a series of joint works with Marco Bertola, Boris Dubrovin and Don Zagier.

 

报告人：虞国富（上海交通大学）

报告题目:  The application of Hirota's bilinear method in the construction of rational and semi-rational solutions of integrable equations

摘要: In this talk, we will first present some review of the application of Hirota's bilinear method in the construction of rational and semi-rational solutions of integrbale equations. In the second part, we investigate a special two-dimensional lattice equation proposed by Blaszak and Szum based on the Hirota's  bilinear method. We derive solitons, breathers and rational solutions to the lattice equation both on the constant and periodic background. These solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. We show that rational solutions are given  in terms of Schur polynomials and demonstrate that these rational solutions  exhibit algebraic solitons and lump solitons. We explore the asymptotic analysis to the algebraic solitons  and  find asymptotic algebraic solitons localized in the straight. 

 

报告人：张成（上海大学）

报告题目:  Aspects of discrete integrable systems

摘要: In this talk, I will first give an overview of some aspects of recent developments in discrete integrable systems. This includes the notion of multi-dimensional consistency, classification results of quad-equations, connections to complex analysis and to some discrete dynamical systems. I will also discuss how to implement discrete initial-boundary value problems for quad-graph systems. 

 

报告人：张国强（中科院数学与系统科学研究院）

报告题目: Rogue waves for vector NLS equation

摘要: In this talk, I will report joint work with Liming Ling and Zhenya Yan on the study of rogue waves. In the framework of the generalized Darboux transformation, explicit rogue wave solutions for the vector NLS equation with any n components are presented. With these explicit formulae, we calculate their peak amplitudes and study the large-parameter asymptotic.

 

报告人：张仑（复旦大学）

报告题目: Gap probability for the hard edge Pearcey process

摘要: The hard edge Pearcey process is universal in random matrix theory and many other stochastic models. In this talk, we consider gap probabilityfor the thinned/unthinned hard edge Pearcey process over the interval (0,s). By working on the relevant Fredholm determinants, we obtain an integral representation of the gap probability via a Hamiltonian related a system of coupled differential equations and the large gap asymptotics. Moreover, we also establish   asymptotic statistical properties of the counting function for the hard edge Pearcey process. Based on joint works with Dan Dai, Shuai-Xia Xu and Luming Yao.