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## Archive of Issues

Uzbekistan Urgench
Year
2022
Volume
32
Issue
2
Pages
153-170
 Section Mathematics Title Integration of the Kaup–Boussinesq system with a self-consistent source via inverse scattering method Author(-s) Babajanov B.A.ab, Azamatov A.Sh.b Affiliations Institute of Mathematics, Khorezm Branch, Uzbekistan Academy of Sciencesa, Urgench State Universityb Abstract 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, Kaup–Boussinesq system, self-consistent source, inverse scattering method, quadratic pencil of Sturm–Liouville equations UDC 517.957 MSC 34L25, 35Q41, 37K10, 35R30, 34M46 DOI 10.35634/vm220201 Received 11 January 2022 Language English Citation Babajanov B.A., Azamatov A.Sh. Integration of the Kaup–Boussinesq system with a self-consistent source via inverse scattering method, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 2, pp. 153-170. References Boussinesq J. 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