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 Nelin. Dinam.: Year: Volume: Issue: Page: Find

 Nelin. Dinam., 2019, Volume 15, Number 1, Pages 21–34 (Mi nd637)

Nonlinear physics and mechanics

Excitation of Large-Amplitude Localized Nonlinear Waves by the Interaction of Kinks of the Sine-Gordon Equation with Attracting Impurity

A. M. Gumerova, E. G. Ekomasovab, R. V. Kudryavtsevac, M. I. Fakhretdinova

a Bashkir State University, ul. Zaki Validi 32, Ufa, 450076 Russia
b South Ural State University (National Research University), pr. Lenina 76, Chelyabinsk, 454080 Russia
c Institute of Molecule and Crystal Physics, pr. Oktyabrya 151, Ufa, 450075 Russia

Abstract: The generation and evolution of localized waves on an impurity in the scattering of a kink of the sine-Gordon equation are studied. It is shown that the problem can be considered as excitation of oscillations of a harmonic oscillator by a short external impulse. The external impulse is modeled by the scattering of a kink on an impurity. The influence of the modes of motion of a kink on the excitation energy of localized waves is numerically and analytically studied. The method of collective coordinate for the analytical study is used. The value of this energy is determined by the ratio of the impurity parameters and the initial kink velocity. It is shown that the dependence of the energy (and amplitude) of the generated localized waves on the initial kink velocity has only one maximum. This behavior is observed for the cases of point and extended impurities. Analytical expression for the amplitude of the localized wave in the case of point impurity is obtained. This allows controlling the excitation energy of localized waves using the initial kink velocity and impurity parameters. The study of the evolution of localized impurities under the action of an external force and damping has shown a good agreement with the nondissipative case. It is shown that small values of the external force have no significant effect on the oscillations of localized waves. An analytical expression for the logarithmic decrement of damping is obtained. This study may help to control the parameters of the excited waves in real physical systems.

Keywords: sine-Gordon equation, impurity, kink, wave generation

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 02.A03.21.0011 Russian Foundation for Basic Research 18-31-00122 This work was supported by Act 211 of the Government of the Russian Federation, contract No. 02.A03.21.0011. For A.M.Gumerov and R.V.Kudryavtsev the work was supported by the RFBR grant, project 18-31-00122.

DOI: https://doi.org/10.20537/nd190103

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MSC: 35C08, 35Q51, 65M06
Accepted:21.03.2019
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Citation: A. M. Gumerov, E. G. Ekomasov, R. V. Kudryavtsev, M. I. Fakhretdinov, “Excitation of Large-Amplitude Localized Nonlinear Waves by the Interaction of Kinks of the Sine-Gordon Equation with Attracting Impurity”, Nelin. Dinam., 15:1 (2019), 21–34

Citation in format AMSBIB
\Bibitem{GumEkoKud19} \by A. M. Gumerov, E. G. Ekomasov, R. V. Kudryavtsev, M. I. Fakhretdinov \paper Excitation of Large-Amplitude Localized Nonlinear Waves by the Interaction of Kinks of the Sine-Gordon Equation with Attracting Impurity \jour Nelin. Dinam. \yr 2019 \vol 15 \issue 1 \pages 21--34 \mathnet{http://mi.mathnet.ru/nd637} \crossref{https://doi.org/10.20537/nd190103} \elib{http://elibrary.ru/item.asp?id=37293019}