AUTHOR=Huang Xiaoqian , Zhang Yong , Dong Huanhe 

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

JOURNAL=Frontiers in Physics

VOLUME=Volume 10 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.1055998

DOI=10.3389/fphy.2022.1055998

ISSN=2296-424X

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