Integration of the Kaup-Boussinesq system with a self-consistent source via inverse scattering method

B. A. Babajanov   ; A. Sh. Azamatov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 2, pp. 153-170

Résumé

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.

Détail

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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