Localized waves of the $\varphi^4$ equation in models with two extended impurities

Fakhretdinov M.I., Ekomasov E.G.
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In this paper, we consider the interaction of a kink of the $\varphi^4$ equation with two identical extended impurities. An extended impurity is described using a rectangular function. The case of an attractive impurity is analyzed. Using analytical methods, we consider the case of small amplitudes of localized waves, when it is possible to linearize the equations of motion. For the numerical solution, the method of lines for partial differential equations was used. To find the oscillation frequencies of waves localized on impurities, the discrete Fourier transform is used. The kink was launched in the direction of the impurities with different initial velocities. The distance between the two impurities was also varied. It is shown that when a kink interacts with impurities, long-lived localized breather-type waves are excited on them. Their structure and coupled dynamics are investigated. It is determined how, by changing the parameters of the impurities and the distance between them, it is possible to control the type and dynamic parameters of the coupled oscillations of the waves localized on the impurities. Possible solutions in the form of in-phase, antiphase oscillations, in the form of beats are found. The oscillations of localized waves occur with the emission of small-amplitude waves. The spectrum of these emissions consists of two frequencies. The first is approximately equal to $\sqrt{2}$, which corresponds to the frequency value for the wobbling breather tail of the $\varphi^4$ equation. The second is approximately equal to the doubled frequency of impurity mode oscillations. The presence of two possible frequencies for coupled localized oscillations is found both analytically and numerically. It is shown that the frequencies strongly depend on the distance between impurities. With increasing distance between impurities, the frequencies merge into one — frequency obtained for the case of a single impurity. The dependences of the frequencies on the distance between impurities found numerically and analytically coincide well for large distances, when the interaction between impurities is weak, and begin to differ noticeably at small distances, when the interaction between impurities is strong. The analytical value of the obtained frequencies is always greater than the numerical ones. It is shown that the dependence of the amplitude of localized waves on the initial kink velocity has several minima and maxima.

Keywords: $\varphi^4$ equation, localized waves, kink, breather, impurity
Citation in English: Fakhretdinov M.I., Ekomasov E.G. Localized waves of the $\varphi^4$ equation in models with two extended impurities // Computer Research and Modeling, 2025, vol. 17, no. 3, pp. 437-449
Citation in English: Fakhretdinov M.I., Ekomasov E.G. Localized waves of the $\varphi^4$ equation in models with two extended impurities // Computer Research and Modeling, 2025, vol. 17, no. 3, pp. 437-449
DOI: 10.20537/2076-7633-2025-17-3-437-449
URL: http://crm-en.ics.org.ru/journal/article/3622/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2025 Fakhretdinov M.I., Ekomasov E.G.

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