可积系统广泛地来自于物理、水波、生物和光纤通讯等应用科学中，具有很多好的性质, 与微分几何中的不变几何流和子流形几何密切相关。本项目首先研究具有非线性色散项的连续可积系统的几何结构如几何可积性及其与不变几何流的密切联系，探讨这些可积系统的可积性质如双Hamiltonian结构、Bäcklund变换和对称等可积性质的几何特征。其次，研究具有非线性色散项的连续可积系统解的奇性的分类、形成的条件和机理，寻求与Camassa-Holm方程不同的爆破机制如曲率爆破等，探讨各种非线性因素和色散效应的相互作用对奇性产生的影响。 探讨孤立波的不稳定性能否引起新的奇性现象，并揭示非光滑孤立波解的不稳定性与可积系统奇性的内在联系。最后研究离散可积系统的椭圆函数解的构造、奇性特征和离散系统的奇点结构与可积性的联系；探讨离散可积系统与不变几何流的关系，进而研究这些离散的不变几何流的奇性及其特征。

Nonlinear integrable systems arise widely from physics, water wave, biology, optic communication and applied sciences etc, and possess a number of nice properties. They are closely related to invariant geometric flows and submanifold geometries. In this project, we first study geometric structures of integrable systems with nonlinear dispersion in a systematic manner. Our focus will be the relationship between the integrable sytems and invariant curve flows or submanifold structures in certain geometries. It is of great interest to explore the geometric formulations of integrability such as bi-Hamiltonian structure, Bäcklund transformation and symmetries. Secondly, we study various structures of singularities for continuous integrable systems with nonlinear dispersion. In particular, we pay more attention to the formation mechanism and classification of singularities to those integrable systems. We will be interested in the singularity phenomena which is different from the case for Camassa-Holm equation. Furthermore, we investigate the effect for singularity formation arising from the interaction among nonlinear terms and nonlinear dispersions. The relationship between the singularities of those integrable systems and their corresponding invariant geometric flows is also explored. Since those systems usually admit non-smooth solitary wave solutions, it is of interest to investigate new singularities arising from instability of non-smooth solitary wave solutions, and study the relationship between the instability of non-smooth solitary wave solutions and singularity of some kind of solutions. Finally, we will study the structures of elliptic function solutons, geometric structures and singularities of discrete integrable systems. In addition, we shall investigate the relationship between discrete integrable systems and discrete geometric flows or submanifold geometries. Research on this project enables us to comprehend the geometric structures, characteristics, classification and formation mechanism of singularities to integrable systems with nonlinear dispersion and discrete integrable systems with certain invariant properties.