报告题目：Vortex pairs in Bose-Einstein condensates: from the quantum Spirograph to symmetric breaking bifurcations

报告人：Ricardo Carretero 教授

报告时间：5月28日 周二 10:30-11:30

报告地点：中北校区数学馆201

主持人：陈勇 教授

报告摘要：

Motivated by recent experiments studying the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates (BECs), we illustrate that such systems can be accurately described by ordinary differential equations (ODEs) incorporating (a) vortex precession induced by the harmonic trap confining the BEC and (b) vortex-vortex interactions. The dynamics is studied in detail at the ODE level, both for the equal and opposite charge vortex pairs. Co-rotating steady states are identified about which perturbations lead to spirographic (epicyclical) motion with excellent agreement with experimental observations. A detailed analysis of the ensuing ODEs reveals the possibility of stable asymmetric states bifurcating from symmetric ones. Cases with more than two vortices are also discussed.

报告人简介：

Dr Ricardo Carretero is a professor in the department of Mathematics San Diego State University. His current main research interests include Applied mathematics, scientific computation, dynamical systems and its applications, Nonlinear wave propagation, nonlinear optics, solitons, localization, breathers, soliton interaction. His NSF-DMS including: Modeling, Analysis, Computation and Experiments of Two-Component Bose-Einstein Condensates. Amount: $300,000.00, period: 08/15/08–08/14/11; Topological Excitations in Bose-Einstein Condensates: Existence, Stability, Dynamics and Interactions. Amount: $91,861.00, period: 07/01/05–06/30/07. He has published one book and over 80 papers including papers published in reputed journals including Phys. Rev. Lett, Phys. Rev. A, Phys. Rev. B, Phys. Rev. E, Physica D.

报告题目: Continuous limits and integrability of a higher order semidiscrete mKdV equation

报告人：朱佐农 教授

报告时间：5月28日13:30-14:30

报告地点：中北校区数学馆201

主持人：陈勇 教授

报告摘要：In this talk, aiming to get more insight on the relations between the higher order semidiscrete mKdV equation and higher order mKdV, we will propose a fifth order semidiscrete mKdV.We not only give its Darboux transformation, explicit solutions and conservation laws, but also show that their continuous limits yield the corresponding results for the fifth order mKdV. This is a joint work with Zhou Tong and He Peng.

个人简介：

朱佐农，上海交通大学数学系教授，博士生导师。1982年本科毕业于东南大学（原南京工学院）数学系，2000年在香港浸会大学数学系获哲学博士学位。学术研究领域是数学物理，研究方向是孤立子和可积系统理论。这一理论的核心问题是研究一大类非线性偏（常）微分方程、非线性微分-差分（差分）方程的可积性。这类非线性方程蕴藏着丰富的数学结构，如可用逆散射方法求解，是Hamiltonian 系统，存在着无穷多个守恒量，存在着多孤子解等，同时在流体力学、等离子体物理、非线性光学、场论等领域有广泛的应用。在连续和离散的可积系统的研究上取得若干重要进展，在有重要影响的国际学术期刊上发表50多篇研究论文。先后主持国家自然科学基金项目4项、上海市浦江人才计划项目1项和教育部留学回国人员基金项目1项。 分别参加香港RGC项目1项和西班牙教育和创新部的科研项目3项。 先后到美国Maryland大学，美国Worcester理工学院，香港浸会大学，西班牙Salamanca大学，西班牙皇后大学，加拿大York 大学，巴西UFPR大学学术访问和工作，开展科研合作研究。

 报告题目: Integrable discretizations and self-adaptive moving mesh methods for a class of nonlinear wave equations

报告人：冯宝峰 教授

报告时间: 5月28日 14:30-15:30

报告地点：中北校区数学馆201

主持人：陈勇 教授

报告摘要：

Recently, much attention has been paid to a class of nonlinear wave equations, which include the Camassa-Holm equation, the Degasperis-Procesi equation and their short-wave limits (the Hunter-Saxton and the reduced Ostrovsky equations), the short pulse and coupled short equations etc. These equations share some common features: (1) they are connected to some well-known integrable systems such as two-dimensional Toda-lattice via hodograph (reciprocal) transformations; (2) they admit bizarre solutions such as loop, cupon, peakon, or breather solutions.

In the present talk, we will report our recent work on integrable discretizations for this class of soliton equations. By Hirota's bilinear method and appropriate discrete Hodograph transformation, we have successfully constructed integrable discretizations for most of these soliton equations, as well as their multi-soliton solutions. In the first part of the talk, we will take a few examples from the list to show how integrable discretizations can be constructed. In the second part of the talk, we will show how these integrable discretizations can be used as a novel numerical scheme: the so-called self-adaptive moving mesh method for the numerical simulation of these nonlinear wave equations. Various numerical experiments including loop, breather and loop-breather interactions will be demonstrated in the presentation. This is a joint work with Dr. Kenichi Maruno, Dr. Yasuhiro Ohta at Kobe University of Japan and Dr. Yong Chen at East China Normal University.

报告人简介：冯宝峰，美国德克萨斯大学泛美分校数学系教授，博士生导师。1989年获得清华大学应用物理及应用数学双学士,1997年日本名古屋大学计算数学硕士，2000年日本京都大学博士，2000年新加坡国立大学计算科学系博士后。多年致力于多方向交叉学科研究：包括偏微分方程的数值解法，可积系统，非线性波动及在流体，固体力学，信息科学及非线性光学方面的应用。近年来着力于对于孤立子方程的离散化以及在保可积结构算法方面的研究。应用方面包括：海洋中的怪波，负折射率的纳米电磁材料的研究。后者在电磁场隐形衣，完全透镜成像方面有着巨大而潜在的应用前景。在Physica D, Phys. Rev. E,J. Phys. A, J. Comput. Phys.等国际刊物上发表论文30多篇，组织过一次国际会议及十多次国际会议的专题。获得过美国能源部（64.5万美元）和国防部（24万美元）的科研基金。

 报告题目：Bäcklund变换、离散系统与可积性

报告人：张大军 教授

报告时间: 5月28日 15:30-16:30

报告地点：中北校区数学馆201

主持人：陈勇 教授

报告摘要：

离散系统及其可积性的研究在过去20年中获得很大进展，推动了离散复分析、离散微分几何等新的数学工具的发展，为目前研究差分方程和离散系统一般理论提供了有效途径。这是一个关于离散系统的入门报告，介绍在离散可积系统研究中的若干基本概念、认识和方法。报告的基本内容如下：（1）如何看待离散，为什么要离散？（2）Bäcklund变换与多维相容性；（3）多维相容性作为可积准则的若干应用“多维相容性”可以定义离散系统的一种可积性。我们将举例说明如何利用多维相容性获得一个离散系统的Bäcklund变换、Lax对以及精确解。

报告人简介：

张大军，上海大学数学系教授，博士生导师。目前主要研究离散可积系统。SIDE (Symmetries and Integrability of Difference Equations) 会议指导委员会委员。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大学非线性科学中心、York大学、Loughborough大学、Glasgow大学、剑桥牛顿数学研究所、美国Texas大学(Pan-American)等，并先后主持国家自然科学基金面上项目3项。