In this paper, we propose and study integrable discrete systems related to the classical Boussinesq system. Based on elementary and binary Darboux transformations and associated Bcklund transformations, both full-discrete systems and semi-discrete systems are constructed. The discrete systems obtained from elementary Darboux transformation are shown to be the discrete systems of relativistic Toda lattice type appeared in the work of Suris (1997) and the ones from binary Darboux transformations are two-component extensions of the lattice potential KdV equation and Kac–van Moerbeke equation. For these discrete systems, their different continuum limits, various interesting reductions and Darboux–Bcklund transformations are considered. Some solutions such as discrete resonant solitons are also presented.

在本文中，我们提出并研究了与经典Boussinesq系统有关的可积分离散系统。基于基本和二进制Darboux转换以及相关的Bcklund转换，可以构建全离散系统和半离散系统。从基本Darboux变换获得的离散系统显示为Suris（1997）工作中出现的相对论Toda晶格类型的离散系统，而从二元Darboux变换获得的离散系统是晶格势KdV方程和Kac– van Moerbeke方程。对于这些离散系统，考虑了它们不同的连续性极限，各种有趣的折减以及Darboux-Bcklund变换。还提出了一些解决方案，例如离散谐振孤子。