ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Ekomasov E. G., Kudryavtsev R. V., Samsonov K. Y., Nazarov V. N., Kabanov D. К. Kink dynamics of the sine-Gordon equation in a model with three identical attracting or repulsive impurities. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 6, pp. 693-709. DOI: 10.18500/0869-6632-003069, EDN: GLJOXX

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.957, 537.611, 51-73
DOI: 
10.18500/0869-6632-003069
EDN: 
GLJOXX

Kink dynamics of the sine-Gordon equation in a model with three identical attracting or repulsive impurities

Autors: 
Ekomasov Evgenii G, Bashkir State University
Kudryavtsev Roman V, Bashkir State University
Samsonov Kirill Yurievich, University of Tyumen
Nazarov Vladimir Nikolaevich,
Kabanov Daniil Константинович,
Abstract: 

Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of the kinks in the sine-Gordon model with “impurities” (or spatial inhomogeneity of the periodic potential).

Methods. Using the method of collective variables for the case of three identical point impurities located at the same distance from each other, a system of differential equations is obtained.

Resulting system of equations makes it possible to describe the dynamics of the kink taking into account the excitation of localized waves on impurities. To analyze the dynamics of the kink in the case of extended impurities, a numerical finite difference method with an explicit integration scheme was applied. Frequency analysis of kink oscillations and localized waves calculated numerically was performed using a discrete Fourier transform.

Results. For the kink dynamics, taking into account the excitation of oscillations in modes, a system of equations for the coordinate of the kink center and the amplitudes of waves localized on impurities is obtained and investigated. Significant differences are observed in the dynamics of the kink when interacting with a repulsive and attractive impurity. The dynamics of the kink in a model with three identical extended impurities, taking into account possible resonant effects, was solved numerically. It is established that the found scenarios of kink dynamics for an extended rectangular impurity are qualitatively similar to the scenarios obtained for a point impurity described using a delta function. All possible scenarios of kink dynamics were determined and described taking into account resonant effects.

Conclusion. The analysis of the influence of system parameters and initial conditions on possible scenarios of kink dynamics is carried out. Critical and resonant kink velocities are found as functions of the impurity parameters.

Key words: 
sine-Gordon equation
kink
soliton
breather
method of collective coordinates
impurity
Acknowledgments: 
This work was supported by Russian Foundation for Basic Research, grant No. 20-31-90048.
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Received: 
15.05.2023
Accepted: 
09.07.2023
Available online: 
17.11.2023
Published: 
30.11.2023
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