Integration of the hierarchy of the Kaup-Boussinesq system via inverse scattering method

In this study we derive a rich hierarchy for the Kaup system with timedependent coefficients in the class of rapidly decreasing functions.We show that a hierarchy of the Kaup-Boussinesq system with time-dependent coefficients is also an important theoretical model, since it is a completely integrable system.For 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 timedependent coefficients 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 time-dependent coefficients.

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