Geodesic
Integration of the~Kaup--Boussinesq system with time-dependent coefficients
Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 63-75

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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.
Mots-clés : Kaup–Boussinesq system, soliton solution.
Keywords: quadratic pencil of Sturm–Liouville operators, inverse scattering method 
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     title = {Integration of {the~Kaup--Boussinesq} system with time-dependent coefficients},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a4/}
}
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B. A. Babajanov; A. Sh. Azamatov; R. B. Atajanova. Integration of the~Kaup--Boussinesq system with time-dependent coefficients. Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 63-75. http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a4/