Integrable discretizations for classical Boussinesq system

Abstract

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 Bäcklund 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–Bäcklund transformations are considered. Some solutions such as discrete resonant solitons are also presented.