可积方程初边值问题解的长时间渐近行为研究

In the 1980s, the Riemann-Hilbert problem, as a more general method than inverse scattering method, began to be applied to the solution and asymptotic analysis of initial / boundary value problems of nonlinear integrable systems. In 1993, Deift, an academician of the American Academy of Sciences, and Zhou, his collaborator, proposed a graceful direct asymptotics analysis method-the nonlinear steepest descent method, to analyze the long-term asymptotics of solution of the initial value of completely integrable systems. In this project, we intend to study the following main contents based on the above problems: (1). Study on the long-time asymptotics behavior of the solutions for iniatial-boundary problem of integrable equations with 2×2 matrix Lax pairs；(2). Study on the long-time asymptotics behavior of the solutions for iniatial-boundary problem of integrable equations with 3×3 matrix Lax pairs. The research of this project not only provides necessary theoretical basis and analytical tools for solving related problems of integrable systems, but also has great significance for the development of integrable systems and differential equations.

上世纪80年代，Riemann-Hilbert问题作为比反散射更一般的方法，开始应用于非线性可积系统求初/边值问题求解和渐进分析。1993年，美国科学院院士Deift及其合作者Zhou提出一种优美的直接渐进分析法-非线性速降法，用于分析反散射完全可积系统初值解的长时间渐进性。本项目基于上述问题拟研究以下主要内容：(1) 与2×2矩阵Lax对有关的可积方程初边值问题解的长时间渐进行为研究；(2)与3×3矩阵Lax对有关的可积方程初边值问题解的长时间渐进行为研究。本项目的研究不仅对可积系统相关问题解决提供必要的理论依据和分析工具，而且对可积系统和微分方程理论发展具有重要意义。

本课题属于天元访问基金，按照课题计划，周口师范学院魏含玉老师在2021年访问了复旦大学数学学院，数学学院在办公室，图书资料等方面给予了支持，魏含玉老师积极参加我的博士研究生讨论班，系统学习和掌握了Riemann-Hilbert方法，并在这领域取得了一系列成果，我们合作完成论文3篇，其中1篇已经发表于重要学术刊物《Nonlinear Dynamics》，后续还有些合作论文将陆续投稿发表，访问取得的效果明显，达到了数学天元访问基金的目的。