在表面浅水波中，Camassa-Holm（CH）方程既可以描述波的破裂现象又具有带奇点的孤立波解。而在内波中,可以描述上述现象的简化模型尚不明确。基于Benjamin-Ono（BO）方程描述了深水波中两层流体间内波演化的物理背景，本项目拟研究具有CH方程结构特征的BO类方程。特别地，该模型含有复合非局部算子的高阶非线性项。申请人将综合利用可积系统、调和分析和偏微分方程的理论和方法，研究目标方程的可积性、适定性以及孤立波解的轨道稳定性。具体地，.(1)构造Lax对、双哈密顿结构，通过建立含奇异积分算子的BO类方程的对偶系统，探讨目标模型的可积性；.(2)利用输运方程理论、函数空间理论以及特征线法，研究该方程波的破裂现象和解的适定性；.(3)发展已有证明光滑/尖峰孤立波稳定性的方法，通过建立含复合非局部算子的孤立波解的分类，讨论单个/多个、光滑/含奇点孤立波解的轨道稳定性。

For shallow water wave in the free surface, the Camassa-Holm（CH）equation admits not only wave breaking phenomena, but also peaked solitary wave solution. However, such kind model have not been discovered yet for internal wave. Since the Benjamin-Ono (BO) equation descries the evolution of internal wave arising from a two fluids system，consider herein is a BO type equation. It is analogous to the CH equation formly and possesses composite higher order nonlocal nonlinearity. In this project, the method for integrable system、harmonic analysis and partial differential equation will be applied to investigate the integrability, well-posedness and orbital stability for solitary wave solution of the BO type equation. More precisely, the PI shall:.(1) consider the Lax-pair，bi-Hamiltonian structure, and construct the dual system of the BO type equation with singular operator to study the integrability;.(2) use the theory on transport equation, functional space, as well as the characteristics to study the wave breaking phenomenon and well-posedness;.(3) develop the method for the orbital stability, classify the solitary wave solutions and discuss their orbital stabilities.