On the Integration of the Hierarchy of the Kaup–Boussinesq System with a Self-Consistent Source

In this article, we consider a hierarchy of the Kaup–Boussinesq system with a self-consistent source. We show that a hierarchy of the Kaup–Boussinesq system with a self-consistent source is also an important theoretical model, since it is a completely integrable system. For a solving the problem under consideration, we use the direct and inverse scattering problem of the Sturm–Liouville equation with an energy-dependent potential. The time evolution of the scattering data for the Sturm–Liouville equation with an energy-dependent potential associated with the solution of the hierarchy of the Kaup–Boussinesq system with a self-consistent source is determined. Using the solution of the inverse scattering problem with respect to the time-dependent scattering data we give an algorithm of constructing the solution of the hierarchy of the Kaup–Boussinesq system with a self-consistent source.