Geodesic
Integration of the Kaup-Boussinesq system with a self-consistent source via inverse scattering method
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 2, pp. 153-170

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In this study we consider the Kaup–Boussinesq system with a self-consistent source. We show that the Kaup–Boussinesq system with a self-consistent source can be integrated by the method of inverse scattering theory. For a 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 potentials associated with the solution of the Kaup–Boussinesq system with a self-consistent source is determined. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Kaup–Boussinesq system with a self-consistent source.
Keywords: nonlinear soliton equation, self-consistent source, inverse scattering method, quadratic pencil of Sturm–Liouville equations.
Mots-clés : Kaup–Boussinesq system 
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     author = {B. A. Babajanov and A. Sh. Azamatov},
     title = {Integration of the {Kaup-Boussinesq} system with a self-consistent source via inverse scattering method},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {153--170},
     publisher = {mathdoc},
     volume = {32},
     number = {2},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VUU_2022_32_2_a0/}
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B. A. Babajanov; A. Sh. Azamatov. Integration of the Kaup-Boussinesq system with a self-consistent source via inverse scattering method. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 2, pp. 153-170. http://geodesic.mathdoc.fr/item/VUU_2022_32_2_a0/