2+1维可积系统研究中的几个问题

可积系统理论是非线性科学的重要组成部分，在数学、物理学和力学中有着重要应用。自从上世纪六十年代以来, 可积系统理论从不同层面都取得飞速发展，特别是关于1+1维的可积系统的研究，与不同的数学分支交织在一起。相比较于1+1维情形，对于2+1维的可积系统的研究要滞后一些，是可积系统理论研究的困难部分之一。本项目主要研究与2+1维可积系统相关的若干问题, 主要包括以下三个方面: 一是探讨2+1维可积系统与无限维Frobenius流形之间联系；二是研究和构造2+1维破裂孤立子型可积系统，并讨论其可积分解和解结构；三是研究2+1维可积系统的几何描述和其相关的代数结构。

The theory of integrable systems is an important part of nonlinear science and has important applications in mathematics, physics and mechanics. Since the 1960s, the theory of integrable systems has made rapid progress from different levels, especially the research on 1+1-dimensional integrable systems, intertwined with different branches of mathematics. Compared to the 1+1-dimensional case, the research on 2+1-dimensional integrable systems is lagging behind, which is one of the difficult parts of theoretical research on integrable systems. This project mainly studies several issues related to 2+1-dimensional integrable systems, mainly including the following three aspects: The first topic is to study the relationship between 2+1-dimensional integrable systems and infinite-dimensional Frobenius manifolds. The second topic is to study how to construct more 2+1-dimensional breaking soliton-type integrable systems, and discuss their solutions and integrable decompostions. The third is to study the geometric description and related algebraic structures of 2+1-dimensional integrable systems.