Journal of Mathematical Physics, Analysis, Geometry
2021, vol. 17, No 2, pp. 233-257   https://doi.org/10.15407/mag17.02.233     ( to contents , go back )
https://doi.org/10.15407/mag17.02.233

On the Construction and Integration of a Hierarchy for the Kaup System with a Self-Consistent Source in the Class of Periodic Functions

A. Yakhshimuratov

Urgench Branch of Tashkent University of Information Technologies named after Muhammad Al-Khwarizmi Al-Khwarizmi street, 110, 220100, Urgench, Uzbekistan
E-mail: albaron@mail.ru

T. Kriecherbauer

Bayreuth University, 95440, Bayreuth, Germany
E-mail: thomas.kriecherbauer@uni-bayreuth.de

B. Babajanov

Urgench State University, H. Alimdjan 14, 220100, Urgench, Uzbekistan
E-mail: a.murod@mail.ru

Received January 28, 2020, revised April 17, 2020.

Abstract

In the paper, we derive a rich hierarchy for the Kaup system with a selfconsistent source in the class of periodic functions. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated quadratic pencil of Sturm-Liouville equations with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy. Moreover, we pick a particular system of the hierarchy and demonstrate the benefits of integrability by proving global existence of solutions for the Cauchy problem and by providing an explicit solution.

Mathematics Subject Classification 2010: 39A23, 35Q51, 34K13, 34K29
Key words: the system of Kaup equations, hierarchy, self-consistent source, quadratic pencil of Sturm-Liouville equations, inverse spectral problem, trace formulas, periodical potential

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