The article published as Early Access!
Kink dynamics in the φ4 model with two extended impurities
The purpose of this study is to use numerical methods to consider the problem of nonlinear kink dynamics for the φ4 equation in a model with two identical extended «impurities» (or spatial inhomogeneity of the potential).
Methods. The φ4 model with inhomogeneities was numerically solved using the method of lines for partial differential equations. The kink was launched in the direction of the inhomogeneities with different initial velocities. The distance between the two impurities was also varied. The kink trajectory after interaction with the impurities was studied. The discrete Fourier transform was used to find the oscillation frequencies of the kink after interaction with spatial inhomogeneities.
Results. The interaction between the kink and two identical extended impurities described by rectangular functions is described. Possible scenarios of kink dynamics are determined, taking into account resonance effects, depending on the magnitude of the system parameters and initial conditions. Critical and resonant velocities of the kink motion are found depending on the impurity parameters and the distance between them. Significant differences are observed in the kink dynamics when interacting with repulsive
and attractive impurities. It is established that among the found scenarios of kink dynamics for the case of extended rectangular impurities, there are scenarios of resonant kink dynamics obtained earlier for the case of one extended impurity, for example, quasi-tunneling and repulsion from an attractive potential.
Conclusion. An 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 and the distance between them.
- Kevrekidis P, Cuevas-Maraver J. A Dynamical Perspective on the 34 Model: Past, Present and Future. Cham: Springer; 2019. 311 p. DOI: 10.1007/978-3-030-11839-6.
- Belova TI, Kudryavtsev AE. Solitons and their interactions in classical field theory. Phys. Usp. 1997;40(4):359–386. DOI: 10.1070/pu1997v040n04abeh000227.
- Schneider T, Stoll E. Molecular-dynamics study of a three-dimensional one-component model for distortive phase transitions. Phys. Rev. B. 1978;17(3):1302–1322. DOI: 10.1103/PhysRevB.17.1302.
- Bishop AR. Defect states in polyacetylene and polydiacetylene. Solid State Communications. 1980;33(9):955–960. DOI: 10.1016/0038-1098(80)90289-6.
- Rice MJ, Mele EJ. Phenomenological theory of soliton formation in lightly-doped polyacetylene. Solid State Communications. 1980;35(6):487–491. DOI: 10.1016/0038-1098(80)90254-9.
- Yamaletdinov RD, Slipko VA, Pershin YV. Kinks and antikinks of buckled graphene: a testing ground for the 34 field model. Phys. Rev. B. 2017;96(9):094306. DOI: 10.1103/PhysRevB.96.094306.
- Yamaletdinov RD, Romanczukiewicz T, Pershin YV. Manipulating graphene kinks through positive and negative radiation pressure effects. Carbon. 2019;141:253–257. DOI: 10.1016/j.carbon.2018.09.032.
- Cuevas-Maraver J, Kevrekidis P, Williams F. The Sine-Gordon Model and Its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics. Cham: Springer; 2014. 263 p. DOI: 10.1007/978-3-319-06722-3.
- Belova TI, Kudryavtsev AE. Quasi-periodic orbits in the scalar classical λ 34 field theory. Physica D: Nonlinear Phenomena. 1988;32(1):18–26. DOI: 10.1016/0167-2789(88)90085-1.
- Marjaneh AM, Saadatmand D, Zhou K, Dmitriev SV, Zomorrodian ME. High energy density in the collision of N kinks in the 34 model. Communications in Nonlinear Science and Numerical Simulation. 2017;49:30–38. DOI: 10.1016/j.cnsns.2017.01.022.
- Takyi I, Weigel H. Collective coordinates in one-dimensional soliton models revisited. Phys. Rev. D. 2016;94(8):085008. DOI: 10.1103/PhysRevD.94.085008.
- Malomed BA. Perturbative analysis of the interaction of a phi4 kink with inhomogeneities. J. Phys. A: Math. Gen. 1992;25(4):755–764. DOI: 10.1088/0305-4470/25/4/015.
- Fei Z, Kivshar YS, Vazquez L. Resonant kink-impurity interactions in the 34 model. Phys. Rev. A. 1992;46(8):5214–5220. DOI: 10.1103/physreva.46.5214.
- Romanczukiewicz T. Creation of kink and antikink pairs forced by radiation. J. Phys. A: Math. Gen. 2006;39(13):3479–3494. DOI: 10.1088/0305-4470/39/13/022.
- Alonso Izquierdo A, Queiroga-Nunes J, Nieto LM. Scattering between wobbling kinks. Phys. Rev. D. 2021;103(4):045003. DOI: 10.1103/PhysRevD.103.045003.
- Ablowitz MJ, Kruskal MD, Ladik JF. Solitary Wave Collisions. SIAM J. Appl. Math. 1979;36(3): 428–437. DOI: 10.1137/0136033.
- Goodman RH, Haberman R. Kink-Antikink Collisions in the phi4 Equation: The n-Bounce Resonance and the Separatrix Map. SIAM J. Appl. Dyn. Syst. 2005;4(4):1195–1228. DOI: 10.1137/050632981.
- Gani VA, Kudryavtsev AE, Lizunova MA. Kink interactions in the (1 + 1)-dimensional 36 model. Phys. Rev. D. 2014;89(12):125009. DOI: 10.1103/PhysRevD.89.125009.
- Marjaneh AM, Saadatmand D, Zhou K, Dmitriev SV, Zomorrodian ME. High energy density in the collision of N kinks in the 34 model. Communications in Nonlinear Science and Numerical Simulation. 2017;49:30–38. DOI: 10.1016/j.cnsns.2017.01.022.
- Yan H, Zhong Y, Liu YX, Maeda K. Kink-antikink collision in a Lorentz-violating 34 model. Phys. Lett. B. 2020;807:135542. DOI: 10.1016/j.physletb.2020.135542.
- Getmanov BS. Bound states of solitons in the 34 2 field theory model. Sov. Phys. JETP Lett.1976;24:291–294.
- Saadatmand D, Dmitriev SV, Borisov DI, Kevrekidis PG, Fatykhov MA, Javidan K. Effect of the34 kink’s internal mode at scattering on a PT-symmetric defect. JETP Lett. 2015;101(7):497–502. DOI: 10.1134/S0021364015070140.
- Saadatmand D, Javidan K. Collective-Coordinate Analysis of Inhomogeneous Nonlinear Klein–Gordon Field Theory. Braz J Phys. 2013;43(1-2):48–56. DOI: 10.1007/s13538-012-0113-y.
- Arash G. Dynamics of 34 Kinks by Using Adomian Decomposition Method. American Journal of Numerical Analysis. 2016;4(1):8–10. DOI: 10.12691/ajna-4-1-2.
- Kalbermann G. Soliton tunneling. Phys. Rev. E. 1997;55(6):R6360–R6362. ¨ DOI: 10.1103/PhysRevE.55.R6360.
- Fakhretdinov MI, Samsonov KY, Dmitriev SV, Ekomasov EG. Kink Dynamics in the 34 Model with Extended Impurity. Rus. J. Nonlin. Dyn. 2023;19(3):303–320. DOI: 10.20537/nd230603.
- Fakhretdinov MI, Samsonov KY, Dmitriev SV, Ekomasov EG. Attractive Impurity as a Generator of Wobbling Kinks and Breathers in the 34 Model. Rus. J. Nonlin. Dyn. 2024;20(1):15–26. DOI: 10.20537/nd231206.
- Ekomasov EG, Samsonov KY, Gumerov AM, Kudryavtsev RV. Nonlinear waves of the sineGordon equation in the model with three attracting impurities. Izvestiya VUZ. Applied Nonlinear Dynamics. 2022;30(6):749–765. DOI: 10.18500/0869-6632-003011.
- Gonzalez JA, Bellorın A, Garсıa-Nustes MA, Guerrero LE, Jimenez S, Vazquez L. Arbitrarilylarge numbers of kink internal modes in inhomogeneous sine-Gordon equations. Phys. Lett. A. 2017;381(24):1995–1998. DOI: 10.1016/j.physleta.2017.03.042.
- Gumerov AM, Ekomasov EG, Murtazin RR, Nazarov VN. Transformation of sine-Gordon solitons in models with variable coefficients and damping. Comput. Math. and Math. Phys. 2015;55(4):628–637. DOI: 10.1134/S096554251504003X.
- Ekomasov EG, Gumerov AM, Murtazin RR. Interaction of sine-Gordon solitons in the model with attracting impurities. Math. Methods Appl. Sci. 2016;40(17):6178–6186. DOI: 10.1002/mma.3908.
- Ekomasov EG, Gumerov AM, Kudryavtsev RV. Resonance dynamics of kinks in the sineGordon model with impurity, external force and damping. Journal of Computational and Applied Mathematics. 2017;312:198–208. DOI: 10.1016/j.cam.2016.04.013.
- Ekomasov EG, Gumerov AM, Kudryavtsev RV, Dmitriev SV, Nazarov VN. Multisoliton Dynamics in the Sine-Gordon Model with Two Point Impurities. Braz. J. Phys. 2018;48(6):576–584. DOI: 10.1007/s13538-018-0606-4.
- Lizunova MA, Kager J, de Lange S, van Wezel J. Kinks and realistic impurity models in 34-theory. Int. J. Mod. Phys. B. 2022;36(05):2250042. DOI: 10.1142/S0217979222500424.
- Ekomasov EG, Kudryavtsev RV, Samsonov KY, Nazarov VN, Kabanov DК. Kink dynamics of the sine-Gordon equation in a model with three identical attracting or repulsive impurities. Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(6):693–709. DOI: 10.18500/0869-6632-003069.
- Schiesser WE. The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press: Elsevier; 2012. 326 p.
- 335 reads