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

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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.

About the authors

Evgenii G Ekomasov

Bashkir State University

ORCID iD: 0000-0002-6194-3358
SPIN-code: 3431-3799
Scopus Author ID: 6507892688
450076, Russian Federation, Republic of Bashkortostan, Ufa, st. Zaki Validi 32

Roman V Kudryavtsev

Bashkir State University

450076, Russian Federation, Republic of Bashkortostan, Ufa, st. Zaki Validi 32

Kirill Yurievich Samsonov

University of Tyumen

ORCID iD: 0000-0002-3170-7872
SPIN-code: 8022-7936
Scopus Author ID: 57216708913
6 Volodarskogo Street, 625003 Tyumen

Vladimir Nikolaevich Nazarov

Институт физики молекул и кристаллов – обособленное структурное подразделение Федерального государственного бюджетного научного учреждения Уфимского федерального исследовательского центра Российской академии наук

ORCID iD: 0000-0002-4749-1367
SPIN-code: 7011-5108
Scopus Author ID: 57198280882
450054, г. Уфа, пр.Октября, 71

Daniil Konstantinovich Kabanov

Уфимский университет науки и технологий

Приволжский федеральный округ, Республика Башкортостан, г. Уфа, ул. Заки Валиди, дом 32

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