Benjamin-Ono(BO)方程描述深水波中的多层流体内引力波的传播，是一类典型的非局部可积系统，其中包含了由Hilbert变换所确定的非局部色散项，该方程得到了多名国际著名数学家的关注和研究。本项目主要研究BO类方程的几何，解的结构及其孤立波的稳定性。首先将研究BO方程、周期的BO方程即中立长波(ILW)方程、描述多种非线性项作用的BO类方程和ILW类方程的几何结构，建立这些可积系统的与某些几何中不变几何流的关系；深入研究这些方程的可积性质，如构造其Lax对，对偶可积系统，Daboux变换和Backlund变换，哈密顿结构和强对称等, 深入探讨BO和ILW类方程的物理背景。其次，发展各种方法研究BO类方程和ILW类方程的多孤立子的轨道稳定性、不稳定性及其渐近稳定性。最后研究BO类方程和ILW类方程解的奇性及其形成，并给出其分类。

The Benjamin-Ono (BO) equation arises in the context of long internal gravity waves in a stratified fluid with finite depth, which is a kind of typical nonlocal integrable system. It contains nonlocal dispersion term involving the Hilbert transformation. That brings much difficult for the study of the BO equation. The BO equation was paid much attention widely by some well-known mathematicians in the world. A number of interesting results were obtained. In this project, we shall study firstly the geometrical structures of the BO equation, intermediate long wave equation, the generalized BO equation and ILW equation descrbing various nonlinearities, and establish the deep connection between the geometrical flows in certain geometries and those integrable equations。 Secondly, we shall study integrable properties including their Lax-pair representation, dual integrable systems, the Daboux transformation and Backlund transformation, Hamiltonian structure and strong symmetries, which is of great interest to examine their physical background. Thirdly, we develop various methods to study orbital stability, instability and asymptotical stability. Finally, we shall discuss formation of singularities, present conditions on the formation of singularities, and provide the classification of singularities.