Excitation of Large-Amplitude Localized Nonlinear Waves by the Interaction of Kinks of the Sine-Gordon Equation with Attracting Impurity

Received 27 January 2019; accepted 21 March 2019

2019, Vol. 15, no. 1, pp. 21-34

Author(s): Gumerov A. M., Ekomasov E. G., Kudryavtsev R. V., Fakhretdinov M. I.

The generation and evolution of localized waves on an impurity in the scattering of a kink of the sine-Gordon equation are studied. It is shown that the problem can be considered as excitation of oscillations of a harmonic oscillator by a short external impulse. The external impulse is modeled by the scattering of a kink on an impurity. The influence of the modes of motion of a kink on the excitation energy of localized waves is numerically and analytically studied. The method of collective coordinate for the analytical study is used. The value of this energy is determined by the ratio of the impurity parameters and the initial kink velocity. It is shown that the dependence of the energy (and amplitude) of the generated localized waves on the initial kink velocity has only one maximum. This behavior is observed for the cases of point and extended impurities. Analytical expression for the amplitude of the localized wave in the case of point impurity is obtained. This allows controlling the excitation energy of localized waves using the initial kink velocity and impurity parameters. The study of the evolution of localized impurities under the action of an external force and damping has shown a good agreement with the nondissipative case. It is shown that small values of the external force have no significant effect on the oscillations of localized waves. An analytical expression for the logarithmic decrement of damping is obtained. This study may help to control the parameters of the excited waves in real physical systems.

Keywords: sine-Gordon equation, impurity, kink, wave generation

Citation: Gumerov A. M., Ekomasov E. G., Kudryavtsev R. V., Fakhretdinov M. I., Excitation of Large-Amplitude Localized Nonlinear Waves by the Interaction of Kinks of the Sine-Gordon Equation with Attracting Impurity, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 21-34

DOI:10.20537/nd190103

Download File
PDF, 475.67 Kb


References

[1]	The Sine-Gordon Model and Its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics, eds. J. Cuevas-Maraver, P. Kevrekidis, F. Williams, Springer, Cham, 2014
[2]	Encyclopedia of Nonlinear Science, ed. A. Scott, Routledge, New York, 2005
[3]	Yakushevich, L. V., Savin, A. V., and Manevitch, L. I., “On the Internal Dynamics of Topological Solitons in DNA”, Phys. Rev. E, 66:1 (2002), 016614, 14 pp.
[4]	Shamsutdinov, M. A., Khabibullin, I. T., Kharisov, A. T., and Tankeyev, A. P., “Dynamics of Magnetic Kinks in Exchange-Coupled Ferromagnetic Layers”, Phys. Metals Metallogr., 108:4 (2009), 327 ; Fiz. Met. i Metalloved., 108:4 (2009), 345–358 (Russian)
[5]	Braun, O. M. and Kivshar, Yu. S., The Frenkel – Kontorova Model: Concepts, Methods, and Applications, Springer, Berlin, 2004, XVIII, 472 pp.
[6]	Dauxois, Th. and Peyrard, M., Physics of Solitons, Cambridge Univ. Press, Cambridge, 2010, xii+422 pp.
[7]	Gani, V. A., Lensky, V., and Lizunova, M. A., “Kink Excitation Spectra in the $(1+1)$-Dimensional $\varphi^8$ Model”, J. High Energ. Phys., 2015:8 (2015), 147, 21
[8]	Gani, V. A., Lizunova, M. A., and Radomskiy, R. V., “Scalar Triplet on a Domain Wall”, J. Phys. Conf. Ser., 675 (2016), 012020, 4 pp.
[9]	Gani, V. A., Kudryavtsev, A. E., and Lizunova, M. A., “Kink Interactions in the $(1+1)$-Dimensional $\varphi^6$ Model”, Phys. Rev. D, 89:12 (2014), 125009, 12 pp.
[10]	Kavitha, L., Parasuraman, E., Gopi, D., Prabhu, A., and Vicencio, R. A., “Nonlinear Nano-Scale Localized Breather Modes in a Discrete Weak Ferromagnetic Spin Lattice”, J. Magn. Magn. Mater., 401 (2016), 394–405
[11]	Kavitha, L., Mohamadou, A., Parasuraman, E., Gopi, D., Akila, N., and Prabhu, A., “Modulational Instability and Nano-Scale Energy Localization in Ferromagnetic Spin Chain with Higher Order Dispersive Interactions”, J. Magn. Magn. Mater., 404 (2016), 91–118
[12]	Golovchan, A. V., Kruglyak, V. V., Tkachenko, V. S., and Kuchko, A. N., “Magnonic Band Spectrum of Spin Waves in an Elliptical Helix”, R. Soc. Open Sci., 5:1 (2018), 172285, 9 pp.
[13]	Askari, A., Saadatmand, D., and Javidan, K., “Collective Coordinate System in $(2+1)$ Dimensions: $CP^1$ Lumps-Potential Interaction”, Waves Random Complex Media, 29:2 (2019), 368–381
[14]	Moradi Marjaneh, A., Askari, A., Saadatmand, D., and Dmitriev, S. V., “Extreme Values of Elastic Strain and Energy in Sine-Gordon Multi-Kink Collisions”, Eur. Phys. J. B, 91:1 (2018), 22, 8 pp.
[15]	Popov, S. P., “Compactons and Riemann Waves of an Extended Modified Korteweg – de Vries Equation with Nonlinear Dispersion”, Comput. Math. Math. Phys., 58:3 (2018), 437–448
[16]	Popov, S. P., “Nonautonomous Soliton Solutions of the Modified Korteweg – de Vries – Sine-Gordon Equation”, Comput. Math. Math. Phys., 56:11 (2016), 1929–1937
[17]	Dubovik, M. N., Korzunin, L. G., and Filippov, B. N., “Asymmetrical Pinning of Vortex Domain Walls in Ferromagnetic Films in Areas with Increased Saturation Magnetization”, Phys. Metals Metallogr., 116:7 (2015), 656–662 ; Fiz. Met. i Metalloved., 116:7 (2015), 694–700 (Russian)
[18]	Ekomasov, E. G., Gumerov, A. M., Kudryavtsev, R. V., Dmitriev, S. V., and Nazarov, V. N., “Multisoliton Dynamics in the Sine-Gordon Model with Two Point Impurities”, Braz. J. Phys., 48:6 (2018), 576–584
[19]	Nonlinear Science at the Dawn of the 21st Century, Lecture Notes in Phys., 542, eds. P. L. Christiansen, M. P. Sørensen, A. C. Scott, Springer, Berlin, 2000
[20]	Currie, J. P., Trullinger, S. E., Bishop, A. R., and Krumhandl, J. A., “Numerical Simulation of Sine-Gordon Soliton Dynamics in the Presence of Perturbations”, Phys. Rev. B, 15:12 (1977), 5567–5580
[21]	Ekomasov, E. G., Gumerov, A. M., and Murtazin, R. R., “Interaction of Sine-Gordon Solitons in the Model with Attracting Impurities”, Math. Models Methods Appl. Sci., 40:17 (2016), 6178–6186
[22]	Ekomasov, E. G., Gumerov, A. M., and Kudryavtsev, R. V., “On the Possibility of the Observation of the Resonance Interaction between Kinks of the Sine-Gordon Equation and Localized Waves in Real Physical Systems”, JETP Lett., 101:12 (2015), 835–839 ; Pis'ma v Zh. Èksper. Teoret. Fiz., 101:12 (2015), 935–939 (Russian)
[23]	Ekomasov, E. G., Gumerov, A. M., and Kudryavtsev, R. V., “Resonance Dynamics of Kinks in the Sine-Gordon Model with Impurity, External Force and Damping”, J. Comput. Appl. Math., 312 (2017), 198–208
[24]	Kivshar, Yu. S., Pelinovsky, D. E., Cretegny, T., and Peyrard, M., “Internal Modes of Solitary Waves”, Phys. Rev. Lett., 80:23 (1998), 5032–5035
[25]	Kivshar, Yu. S., Malomed, B. A., Zhang, F., and Vazquez, L., “Creation of Sine-Gordon Solitons by a Pulse Force”, Phys. Rev. B, 43:1 (1991), 1098–1109
[26]	González, J. A., Bellorín, A., and Guerrero, L. E., “Internal Modes of Sine-Gordon Solitons in the Presence of Spatiotemporal Perturbations”, Phys. Rev. E (3), 65:6 (2002), 065601, 4 pp.
[27]	Kivshar, Yu. S. and Malomed, B. A., “Addendum: Dynamics of Solitons in Nearly Integrable Systems”, Rev. Mod. Phys., 63:1 (1991), 211–212
[28]	Javidan, K., “Analytical Formulation for Soliton-Potential Dynamics”, Phys. Rev. E, 78:4 (2008), 046607, 8 pp.
[29]	Chacón, R., Bellorín, A., Guerrero, L. E., and González, J. A., “Spatiotemporal Chaos in Sine-Gordon Systems Subjected to Wave Fields: Onset and Suppression”, Phys. Rev. E, 77:4 (2008), 046212, 4 pp.
[30]	González, J. A., Bellorín, A., Reyes, L. I., Vásquez, C., and Guerrero, L. E., “Geometrical Resonance in Spatiotemporal Systems”, Europhys. Lett., 64:6 (2003), 743–749
[31]	González, J. A., Cuenda, S., and Sánchez, A., “Kink Dynamics in Spatially Inhomogeneous Media: The Role of Internal Modes”, Phys. Rev. E (3), 75:3 (2007), 036611, 7 pp.
[32]	Gumerov, A. M., Ekomasov, E. G., Murtazin, R. R., and Nazarov, V. N., “Transformation of sine-Gordon solitons in models with variable coefficients and damping”, Comput. Math. Math. Phys., 55:4 (2015), 628–637
[33]	González, J. A., Bellorín, A., García-Ñustes, M. A., Guerrero, L. E., Jiménez, S., and Vázquez, L., “Arbitrarily Large Numbers of Kink Internal Modes in Inhomogeneous Sine-Gordon Equations”, Phys. Lett. A, 381:24 (2017), 1995–1998
[34]	González, J. A. Jiménez, S., Bellorín, A., Guerrero, L. E., and Vázquez, L., “Internal Degrees of Freedom, Long-Range Interactions and Nonlocal Effects in Perturbed Klein – Gordon Equations”, Phys. A, 391:3 (2012), 515–527
[35]	Saadatmand, D., Dmitriev, S. V., Borisov, D. I., and Kevrekidis, P. G., “Interaction of Sine-Gordon Kinks and Breathers with a Parity-Time-Symmetric Defect”, Phys. Rev. E, 90:5 (2014), 052902, 10 pp.
[36]	Belova, T. I. and Kudryavtsev, A. E., “Solitons and Their Interactions in Classical Field Theory”, Physics-Uspekhi, 40:4 (1997), 359–386 ; Uspekhi Fiz. Nauk, 167:4 (1997), 377–406 (Russian)
[37]	Popov, S. P., “Influence of Dislocations on Kink Solutions of the Double Sine-Gordon Equation”, Comput. Math. Math. Phys., 53:12 (2013), 1891–1899
[38]	Malomed, B. A., “Dynamics of Quasi-One-Dimensional Kinks in the Two-Dimensional Sine-Gordon Model”, Phys. D, 52:2–3 (1991), 157–170
[39]	Saadatmand, D. and Javidan, K., “Collective-Coordinate Analysis of Inhomogeneous Nonlinear Klein – Gordon Field Theory”, Braz. J. Phys., 43:1–2 (2013), 48–56
[40]	Ekomasov, E. G., Gumerov, A. M., Murtazin, R. R., Kudryavtsev, R. V., Ekomasov, A. E., and Abakumova, N. N., “Resonant Dynamics of the Domain Walls in Multilayer Ferromagnetic Structure”, Solid State Phenom., 233–234 (2015), 51–54
[41]	Ekomasov, E. G. and Shabalin, M. A., “Simulation the Nonlinear Dynamics of Domain Walls in Weak Ferromagnets”, Phys. Metals Metallogr., 101:Suppl. 1 (2006), S48–S50
[42]	Ekomasov, E. G., Murtazin, R. R., Bogomazova, O. B., and Gumerov, A. M., “One-Dimensional Dynamics of Domain Walls in Two-Layer Ferromagnet Structure with Different Parameters of Magnetic Anisotropy and Exchange”, J. Magn. Magn. Mater., 339 (2013), 133–137
[43]	Ekomasov, E. G., Murtazin, R. R., Bogomazova, O. B., and Nazarov, V. N., “Excitation and Dynamics of Domain Walls in Three-Layer Ferromagnetic Structure with Different Parameters of the Magnetic Anisotropy and Exchange”, Mater. Sci. Forum, 845 (2016), 195–198
[44]	Gulevich, D. R. and Kusmartsev, F. V., “Perturbation Theory for Localized Solutions of the Sine-Gordon Equation: Decay of a Breather and Pinning by a Microresistor”, Phys. Rev. B, 74:21 (2006), 214303, 5 pp.
[45]	Gumerov, A. M., Ekomasov, E. G., Kudryavtsev, R. V., and Fakhretdinov, M. I., “Localized Magnetic Inhomogeneities Generation on Defects As a New Channel of Damping for a Moving Domain Wall”, Letters on Materials, 8:3 (2018), 299–304 (Russian)
[46]	Paul, D. I., “Soliton Theory and the Dynamics of a Ferromagnetic Domain Wall”, J. Phys. C, 12:3 (1979), 585–593
[47]	Piette, B. and Zakrzewski, W. J., “Scattering of Sine-Gordon Kinks on Potential Wells”, J. Phys. A, 40:22 (2007), 5995–6010
[48]	Gumerov, A. M. and Ekomasov, E. G., “Study of the Effect of Point Defects on the Nonlinear Dynamics of Magnetic Nonuniformities”, Letters on Materials, 3:2 (2013), 103–105 (Russian)
[49]	Ekomasov, E. G., Gumerov, A. M., and Kudryavtsev, R. V., “Dynamics of Localized Magnetic Inhomogeneities in the Five-Layer Ferromagnetic Structure”, Letters on Materials, 6:2 (2016), 138–140 (Russian)
[50]	Ekomasov, E. G., Azamatov, S. A., and Murtazin, R. R., “Studying the Nucleation and Evolution of Magnetic Inhomogeneities of the Soliton and Breather Type in Magnetic Materials with Local Inhomogeneities of Anisotropy”, Phys. Metals Metallogr., 105:4 (2008), 313–321 ; Fiz. Met. i Metalloved., 105:4 (2008), 341–349 (Russian)
[51]	Zhang, F., Kivshar, Yu. S., and Vazquez, L., “Resonant Kink-Impurity Interactions in the Sine-Gordon Model”, Phys. Rev. A, 45:8 (1992), 6019–6030
[52]	Landa, P. S., Nonlinear Oscillations and Waves in Dynamical Systems, Math. Appl., 360, Springer, Dordrecht, 2013
[53]	Goodman, R. H., Holmes, P. J., and Weinstein, M. I., “Interaction of Sine-Gordon Kinks with Defects: Phase Space Transport in a Two-Mode Model”, Phys. D, 161:1 (2002), 21–44
[54]	Ekomasov, E. G., Murtazin, R. R., and Nazarov, V. N., “Excitation of Magnetic Inhomogeneities in Three-Layer Ferromagnetic Structure with Different Parameters of the Magnetic Anisotropy and Exchange”, J. Magn. Magn. Mater., 385 (2015), 217–221

This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License

Excitation of Large-Amplitude Localized Nonlinear Waves by the Interaction of Kinks of the Sine-Gordon Equation with Attracting Impurity

References