Integration of the Kaup–Boussinesq system with time-dependent coefficients

We consider the Kaup–Boussinesq system with time-dependent coefficients. We show that the Kaup–Boussinesq system with an additional term is also an important theoretical model, since it is a completely integrable system. We find the time evolution of scattering data for a quadratic pencil of Sturm–Liouville operators associated with the solution of the Kaup–Boussinesq system with time-dependent coefficients. The resulting equalities completely determine the scattering data at any $t$, which allows using the inverse scattering method for solving the Cauchy problem for the Kaup–Boussinesq system with time-dependent coefficients. An example is given to illustrate the application of the obtained results.