Integration of the Kaup–Boussinesq system with a self-consistent source via inverse scattering method

In this study we consider the Kaup–Boussinesq system with a self-consistent source. We show that source can be integrated by method of inverse scattering theory. For solving problem under consideration, use direct and Sturm–Liouville equation an energy-dependent potential. The time evolution data for potentials associated solution is determined. obtained equalities completely determine any $t$, which makes it possible to apply solve Cauchy