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
ABI
Abstract
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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