Localized nonlinear waves of the sine-Gordon equation in a model with three extended impurities

Samsonov K.Y., Kabanov D.K., Nazarov V.N., Ekomasov E.G.
 pdf (3596K)

In this work, we use analytical and numerical methods to consider the problem of the structure and dynamics of coupled localized nonlinear waves in the sine-Gordon model with three identical attractive extended “impurities”, which are modeled by spatial inhomogeneity of the periodic potential. Two possible types of coupled nonlinear localized waves are found: breather and soliton. The influence of system parameters and initial conditions on the structure, amplitude, and frequency of localized waves was analyzed. Associated oscillations of localized waves of the breather type as in the case of point impurities, are the sum of three harmonic oscillations: in-phase, in-phase-antiphase and antiphase type. Frequency analysis of impurity-localized waves that were obtained during a numerical experiment was performed using discrete Fourier transform. To analyze localized breather-type waves, the numerical finite difference method was used. To carry out a qualitative analysis of the obtained numerical results, the problem was solved analytically for the case of small amplitudes of oscillations localized on impurities. It is shown that, for certain impurity parameters (depth and width), it is possible to obtain localized solitontype waves. The ranges of values of the system parameters in which localized waves of a certain type exist, as well as the region of transition from breather to soliton types of oscillations, have been found. The values of the depth and width of the impurity at which a transition from the breather to the soliton type of localized oscillations is observed were determined. Various scenarios of soliton-type oscillations with negative and positive amplitude values for all three impurities, as well as mixed cases, were obtained and considered. It is shown that in the case when the distance between impurities much less than one, there is no transition region where which the nascent breather, after losing energy through radiation, transforms into a soliton. It is shown that the considered model can be used, for example, to describe the dynamics of magnetization waves in multilayer magnets.

Keywords: sine-Gordon equation, kink, breather, soliton
Citation in English: Samsonov K.Y., Kabanov D.K., Nazarov V.N., Ekomasov E.G. Localized nonlinear waves of the sine-Gordon equation in a model with three extended impurities // Computer Research and Modeling, 2024, vol. 16, no. 4, pp. 855-868
Citation in English: Samsonov K.Y., Kabanov D.K., Nazarov V.N., Ekomasov E.G. Localized nonlinear waves of the sine-Gordon equation in a model with three extended impurities // Computer Research and Modeling, 2024, vol. 16, no. 4, pp. 855-868
DOI: 10.20537/2076-7633-2024-16-4-855-868
URL: http://crm-en.ics.org.ru/journal/article/3497/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Indexed in Scopus

Full-text version of the journal is also available on the web site of the scientific electronic library eLIBRARY.RU

The journal is included in the Russian Science Citation Index

The journal is included in the RSCI

International Interdisciplinary Conference "Mathematics. Computing. Education"

Copyright © 2009–2024 Institute of Computer Science