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 Ufimsk. Mat. Zh., 2020, Volume 12, Issue 1, Pages 104–114 (Mi ufa506)

Integration of equations of Kaup system kind with self-consistent source in class of periodic functions

A. B. Yakhshimuratova, B. A. Babajanovb

a Urgench Branch of Tashkent University of Information Technologies named after Muhammad al-Khwarizmi, Al-Khwarizmi street, 110, 220100, Urgench city, Uzbekistan
b Urgench State University, Hamid Alimjan street, 14, 220100, Urgench city, Uzbekistan

Abstract: In this paper, we consider the equations of Kaup system kind with a self-consistent source in the class of periodic functions. We discuss the complete integrability of the considered nonlinear system of equations, which 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 equations of Kaup system kind with self-consistent source in the class of periodic functions. Moreover, it is shown that spectrum of the quadratic pencil of Sturm–Liouville equations with periodic coefficients associated with considering nonlinear system does not depend on time. In a one-gap case, we write the explicit formulae for solutions of the problem under consideration expressed in terms of the Jacobi elliptic functions. We show that if $p_{0} (x)$ and $q_{0} (x)$ are real analytical functions, the lengths of the gaps corresponding to these coefficients decrease exponentially. The gaps corresponding to the coefficients $p(x,t)$ and $q(x,t)$ are same. This implies that the solutions of considered problem $p(x,t)$ and $q(x,t)$ are real analytical functions in $x$.

Keywords: equations of Kaup system kind, quadratic pencil of Sturm–Liouville equations, inverse spectral problem, trace formulas, periodical potential.

 Funding Agency Grant Number Keele University International Erasmus+Program KA106-2 The authors express their gratitude to Prof. Aknazar Khasanov (Samarkand State University, Uzbekistan) for discussion and valuable advice, as well as to the International Erasmus+Program KA106-2, Keele University, UK.

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English version:
Ufa Mathematical Journal, 2020, 12:1, 103–113 (PDF, 431 kB); https://doi.org/10.13108/2020-12-1-103

Bibliographic databases:

UDC: 517.957
MSC: 39A23, 35Q51, 34K13, 34K29
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Citation: A. B. Yakhshimuratov, B. A. Babajanov, “Integration of equations of Kaup system kind with self-consistent source in class of periodic functions”, Ufimsk. Mat. Zh., 12:1 (2020), 104–114; Ufa Math. J., 12:1 (2020), 103–113

Citation in format AMSBIB
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