Geodesic
Nonlinear waves of the sine-Gordon equation in the model with three attracting impurities
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 30 (2022) no. 6, pp. 749-765

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Purpose of this work is to 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 impurities (or spatial inhomogeneity of the periodic potential). Methods. Using the analytical method of collective coordinates for the case of the arbitrary number the same point impurities on the same distance each other, differential equation system was got for localized waves amplitudes as the functions on time. We used the finite difference method with explicit scheme for the numerical solution of the modified sine-Gordon equation. We used a discrete Fourier transform to perform a frequency analysis of the oscillations of localized waves calculate numerically. Results. We found of the differential equation system for three harmonic oscillators with the elastic connection for describe related oscillations of nonlinear waves localized on the three same impurity. The solutions obtained from this system of equations for the frequencies of related oscillation well approximate the results of direct numerical modeling of a nonlinear system. Conclusion. In the article shows that the related oscillation of nonlinear waves localized on three identical impurities located at the same distance from each other represent the sum of three harmonic oscillations: in-phase, in-phase-antiphase and antiphase type. The analysis of the influence of system parameters and initial conditions on the frequency and type of associated oscillations is carried out.
Keywords: sine-Gordon equation, kink, soliton, breather, the method of collective coordinates, impurity. 
@article{IVP_2022_30_6_a6,
     author = {E. G. Ekomasov and K. Yu. Samsonov and A. M. Gumerov and R. V. Kudryavtsev},
     title = {Nonlinear waves of the {sine-Gordon} equation in the model with three attracting impurities},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {749--765},
     publisher = {mathdoc},
     volume = {30},
     number = {6},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2022_30_6_a6/}
}
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E. G. Ekomasov; K. Yu. Samsonov; A. M. Gumerov; R. V. Kudryavtsev. Nonlinear waves of the sine-Gordon equation in the model with three attracting impurities. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 30 (2022) no. 6, pp. 749-765. http://geodesic.mathdoc.fr/item/IVP_2022_30_6_a6/