前沿论坛与短期课程

国家天元数学西北中心定于3月31日至4月9日在腾讯会议室线上举办“孤立子和可积系统前沿理论讲习班”的第一期活动。该讲习班是西北中心2022年促进西部地区发展活动之一，主要为从事相关研究的青年教师及在校研究生系统性的讲授孤立子和可积系统理论中的一些基本理论和核心方法，促进西北及周边地区教师人才在基础数学方面研究质量的提升。本次讲习班第一期活动的课程内容主要涉及KP理论及其应用，包括六次课程（每次两个小时）和两个前沿学术讲座。

一、课程安排

时间：3月31日-4月9日

地点：腾讯会议497 9327 3214（无密码）

二、课程简介

本期讲习班活动分为六次课程（每次两个小时）和两个前沿学术讲座，全程在线上举办。系列课程的主题是“KP theory and applications”，每次课程的简介如下：

Class 1: Introduction to Integrable Systems.

The concept and some important examples of continuous and discrete integrable systems will be introduced. These examples will include KdV, modified KdV, sine-Gordon, nonlinear Schrödinger (NLS), Yajima-Oikawa, derivative NLS equations, Fokas-Lenells equation and massive-Thirring model.

Class 2: Discrete KP and Discrete Two-dimensional Toda-Lattice.

We will give a review on discrete KP (dKP) and discrete two-dimensional Toda-lattice (d2DTL) hierarchies. First, we present bilinear form, Lax pair, Backlund transformation and tau functions of the dKP hierarchy. Then, we connect dKP with d2DTL by reparametrization. In the last, the geometric configuration of the dKP will be reviewed.

Class 3: From dKP to KP, mKP and d2DTL.

In this talk, we will discuss how to generate KP, modified KP and 2DTL hierarchies by Miwa transformation. Starting from dKP equation, it is shown that the whole KP and mKP hierarchy can be generated. Furthermore, discrete 2DTL hierarchy can also be generated. The Backlund transformation will be also deduced.

Class 4: KP Reduction (I).

KP reduction technique is an important method in constructing various solutions of soliton equations. In this talk, we will derive KdV, Boussinesq, NLS equations from KP hierarchy.

Class 5: KP Reduction (II).

We continue discuss the KP reduction for discrete and continuous 2DTL hierarchy. We will first show how can we derive Yajima-Oikawa, Fokas-Lenells and massive-Thirring model equations can be derived from the same set of bilinear equations.

Class 6: Dark solitons, Breathers and Rogue Waves.

We will present a framework in constructing various soliton solutions to discrete and continuous soliton equations under non-vanishing boundary condition. We will show a systematic method to construct from dark to breather and rogue wave solutions via Hirota’s bilinear approach.

前沿讲座一：Various soliton solutions to the massive Thirring model.

Abstract：In this talk, we firstly bilinearize the massive Thirring (MT) model under both the zero and nonzero boundary conditions. Then through the KP hierarchy reduction method, we construct the general bright and dark soliton solutions to the MT model. Finally, we also construct multi-breather solution to the MT model. The connection of the MT model and other soliton equations such as the derivative NLS, the complex short pulse equation, Fokas-Lenells equations will also be discussed. This is a joint work with Dr. Junchao Chen at Lishui University.

前沿讲座二：The Multi-hump and Multi-valley Solitons in The Integrable Models.

Abstract：In the integrable models, we believe that the single solitons usually admit the structure of single hump or single valley. In this talk, we would like to introduce the multi-hump and multi-valley solitons to the related integrable equations based on the analysis of determinant solutions. Moreover, we provide the analysis for the interactions between these multi-hump or multi-valley solitons, which discloses that their interactions are inelastic rather than elastic in general. These results will further deepen the understanding for the phenomenon of solitons in the integrable models.

三、专家简介：

Bao-feng Feng is currently a Professor at the School of Mathematics and Statistical Science of the University of Texas Rio Grande Valley. His research interests focus on applied mathematics especially in nonlinear science. He is a well-known scholar on integrable systems and nonlinear waves. He has published nearly 100 papers and received eight major research grants with more than one million U.S. dollars from multiple research agencies such as U.S. National Science Foundation (NSF), U.S. Department of Defense (DoD) and National Natural Science Foundation of China (NSFC). He delivered a 45 minutes invited talk at International Congress of Chinese Mathematicians in 2019. He currently serves as the Editor for the journal of Physica D.

凌黎明，华南理工大学数学学院教授，博士生导师。主要从事非线性可积系统的研究，在可积系统“怪波”理论的发展中作出了一系列工作，率先同合作者给出高阶怪波解的 Darboux 变换方法以及无穷阶怪波的分析理论。 报告人在该方向上发表 40余篇 SCI 论文，合作出版专著一部。 已发表文章在 Google 学术搜索统计引用 2600 余次，其中单篇最高引用 600 余次，4篇入选ESI高被引论文。主持国家自然科学基金项目3项(含一项优青)。

四、参加方式

讲习班不收取任何费用，从事孤立子和可积系统理论的青年教师和研究生均可在线参加。为便于统计学员信息，请拟参加的学者于3月31日前扫描下方二维码或通过链接填写学员信息表：https://docs.qq.com/form/page/DZXFHVkdaRkdEeFpQ。

五、联系人

刘小川 教授 西安交通大学

Email：liuxiaochuan@mail.xjtu.edu.cn

白佳宁 国家天元数学西北中心

Email：xbty@xjtu.edu.cn