Radiation of breathers in the Klein – Gordon model with impurities

Ekomasov E.G., Kabanov D.K., Fakhretdinov M.I.
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Using numerical methods, the spatiotemporal structure and spectral properties of radiation of long-lived nonlinear localized breather-type waves have been investigated for two nonlinear differential Klein – Gordontype equations — the sine-Gordon equation and the ϕ4 equation — in the presence of a one-dimensional spatial modulation of the potential acting as an attractive impurity. The case of an extended impurity with a rectangular profile, which acts as a potential well for a moving topological soliton (kink) and a generator of localized oscillations, is considered. To solve the initial-boundary value problems, the finite difference method and the method of lines were applied. To increase computational performance and reduce simulation time, vector optimization of the numerical algorithms in Python was implemented using the NumPy library, which made it possible to maintain the required accuracy of calculations over long time intervals. It is established that the localized waves excited after the passage of the impurity are accompanied by a gradual decay of their amplitude due to energy loss to wave radiation. The time dependence of this radiation is divided into zones of strong and weak nonlinearity. In the former, the radiation of the transmitted kink is superimposed on the radiation of the excited breather. Fourier spectral analysis of the radiated waves revealed the presence of two main frequencies at all investigated observation points: the first corresponds to the lower bound of the linear phonon spectrum ($\omega \approx 1$ for the sine-Gordon and $\omega \approx \sqrt{2}$ for the $\varphi^4$), and the second is related to the natural frequency of the breather oscillations. For the sine-Gordon equation, the second frequency is triple the breather oscillation frequency, whereas for the $\varphi^4$ equation, it is double the frequency. It is revealed that with increasing distance from the impurity, the contribution of the phonon spectrum frequency becomes dominant. It is shown that the radiation amplitude slowly decreases with time, following the decrease in the amplitude of the breather itself, and its absolute value far from the impurity for the case of the sine-Gordon equation significantly exceeds the corresponding value for the $\varphi^4$ equation. The obtained quantitative relationships show that by varying the impurity parameters, it is possible to modify the characteristics (frequency spectrum and amplitude) of the waves radiated by the breather.

Keywords: kink, breather, sine-Gordon equation, $\varphi^4$ equation, impurity
Citation in English: Ekomasov E.G., Kabanov D.K., Fakhretdinov M.I. Radiation of breathers in the Klein – Gordon model with impurities // Computer Research and Modeling, 2026, vol. 18, no. 5, pp. 1189-1203
Citation in English: Ekomasov E.G., Kabanov D.K., Fakhretdinov M.I. Radiation of breathers in the Klein – Gordon model with impurities // Computer Research and Modeling, 2026, vol. 18, no. 5, pp. 1189-1203
DOI: 10.20537/2076-7633-2026-18-5-1189-1203
URL: http://crm-en.ics.org.ru/journal/article/3776/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Ekomasov E.G., Kabanov D.K., Fakhretdinov M.I.

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