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 Model. Anal. Inform. Sist., 2016, Volume 23, Number 5, Pages 635–656 (Mi mais529)

Asymptotic integration of a certain second-order linear delay differential equation

P. N. Nesterov

P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia

Abstract: We construct some asymptotic formulas for solutions of a certain linear second-order delay differential equation when the independent variable tends to infinity. Two features concerning the considered equation should be emphasized. First, the coefficient of this equation has an oscillatory decreasing form. Second, when the delay equals zero, this equation turns into the so-called one-dimensional Schrödinger equation at energy zero with Wigner–von Neumann type potential. Dynamics of the latter is well-known. The question of interest is how the behavior of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. This equation also attracts interest from the standpoint of the theory of oscillations of solutions of functional differential equations. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients. The essence of the method is to construct a so-called critical manifold in the phase space of the considered dynamical system. This manifold is attractive and positively invariant, and, therefore, the dynamics of all solutions of the initial equation is determined by the dynamics of the solutions lying on the critical manifold. The system that describes the dynamics of the solutions lying on the critical manifold is a linear system of two ordinary differential equations. To construct the asymptotics for solutions of this system, we use the averaging changes of variables and transformations that diagonalize variable matrices. We reduce the system on the critical manifold to what is called the $L$-diagonal form. The asymptotics of the fundamental matrix of $L$-diagonal system may be constructed by the use of the classical Levinson's theorem.

Keywords: asymptotics, delay differential equation, Shrödinger equation, oscillating coefficients, oscillations of solutions, Levinson's theorem, method of averaging.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation МК-4625.2016.1 This research was supported by the grant of the President of the Russian Federation No. MK-4625.2016.1.

DOI: https://doi.org/10.18255/1818-1015-2016-5-635-656

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UDC: 517.929

Citation: P. N. Nesterov, “Asymptotic integration of a certain second-order linear delay differential equation”, Model. Anal. Inform. Sist., 23:5 (2016), 635–656

Citation in format AMSBIB
\Bibitem{Nes16} \by P.~N.~Nesterov \paper Asymptotic integration of a certain second-order linear delay differential equation \jour Model. Anal. Inform. Sist. \yr 2016 \vol 23 \issue 5 \pages 635--656 \mathnet{http://mi.mathnet.ru/mais529} \crossref{https://doi.org/10.18255/1818-1015-2016-5-635-656} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3569859} \elib{http://elibrary.ru/item.asp?id=27202312}