Integration of the Kaup–Boussinesq system with time-dependent coefficients
Bazar BabajanovRomanovsky Institute of Mathematics, Khorezm Branch of the Academy of Sciences of Uzbekistan, Urgench, UzbekistanA. Sh. AzamatovUrgench State University, Urgench, UzbekistanRukhiya Babazhanovna AtajanovaUrgench State University, Urgench, Uzbekistan
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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.
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