可积系统初边值问题解的长时间渐近分析

In 1993，the nonlinear steepest descent method was first developed by Deift and zhou to determine the asymptotics of oscillatory Riemann-Hilbert problem, where the long-time behavior of the classical solution of modified KdV equation on the line was established. In 1997, Fokas extended Riemann-Hilbert method to initial-boundary volue problems. How to use this to analyze long-time behavior of the classical solution of initial-boundary value of integrable systems is one of open sixteen problems presented by Deift in 2007. This open problem was partly break throughed by yong mathematician Lenells in 2015. The long-time asymptotics of solution for initial boundary problem of integrable systems is a new and frontier field. This will project concentrate on the following two problems related to Deift's open problems: 1. Long-time asymptotics for initial boundary problem of integrable systems with two order matrix Lax pairs on half-line and interval. 2. Long-time asymptotics for initial boundary problem of integrable systems with higher order matrix Lax pairs on half-line and interval. 3. Under innitial data in weight Sobolev space, Long-time asymptotics for initial boundary problem of integrable systems . The results of this project will contribute the study of properties, structure and inner connnections of integrable systems.

1993年，纽约大学Deift院士及合作者Zhou发展了分析可积系统初值问题严格经典解长时间渐近性的Riemann-Hilbert（RH）方法/非线性速降法。1997年，剑桥大学Fokas将RH方法应用到可积系统的初边值问题，但如何分析可积系统初边值问题解的长时间渐近性，这是Deift在2007年所提出的重要公开问题之一。2015年，瑞典皇家理工青年数学家Lenells在这一问题获得部分突破，形成了现今全新的前沿研究方向。为此，本课题打算在这方面重点研究：1. 具有二阶矩阵Lax对可积系统初边值问题解的长时间渐近性。2. 具有三阶矩阵Lax对可积系统初边值问题解的长时间渐近性。3. 在加权Sobolev空间的初始数据和边值下，可积系统初边值问题解的长时间渐近性。本课题是我们前一个基金课题的延续和纵深发展，研究成果将为揭示可积系统的性质、结构和内在联系提供新的结果和方法。

按照课题计划，在可积系统初边值问题求解和渐近分析方面，我们课题组开展了深入研究，取得了一批重要成果，在《J. Diff. Equs.》、《J Nonl.Sci.》、《Stud Appl Math.》等国外重要学术刊物发表SCI 论文33篇，出站博士后3名、培养毕业博士4 名、硕士6 名；在读博士14 名、博士后1 名。邀请30 名国内外学者访问复旦，并作学术报告，课题组成员出境访问学术交流6次，举办学术会议3 次，国内外应邀作学术报告30 余次，超额完成了预期目标