The coupled Boussinesq equation and its Darboux transformation on the time–space scale

Gel’fand-Dikii (GD) formalism is an approach for generating integrable systems in terms of fractional powers of the δ differential operator. In this paper, it extends the GD formalism associated with the third-order δ differential operator L to the time scale. Then, the coupled Boussinesq equation on the time–space scale is given by taking special values, and it can be reduced on different time–space scales. Moreover, the exact solutions of the coupled Boussinesq equation on the time–space scale and the classical Boussinesq equation are constructed via employing the extensions of the Darboux theorem and Crum theorem on the time scale.