On spatio-temporal dynamics of sine-Gordon soliton in nonlinear non-homogeneous media using fully implicit spectral element scheme

References

• Scoott AC. A nonlinear Klein-Gordon equation. Am J Phys. 1969;37(1):52–61. doi: 10.1119/1.1975404
   Web of Science ®Google Scholar
• Coleman S. Quantum Sine-Gordon equation as the massive Thirring model. Phys Rev D. 1975;11(8):2088–2097. doi: 10.1103/PhysRevD.11.2088
   Web of Science ®Google Scholar
• Frenkel J, Kontorova T. On the theory of plastic deformation and twinning. Fiz Zhurnal. 1939;1:137–49.
   Google Scholar
• Gpmzalex JA, Martin-Landrove M. Solitons in a nonlinear DNA model. Phys Lett A. 1994;191:409–415. doi: 10.1016/0375-9601(94)90795-1
   Web of Science ®Google Scholar
• Yakushevich LV. Nonliear physics of DNA. Berlin: Wiley-VCH; 2004.
   Google Scholar
• Gibbon JD, James IN, Moroz IM. An example of soliton behavior in a rotating baroclinic fluid. Proc Roy Soc A. 1979;367:219–237.
   Google Scholar
• Saravanan M, Jagtap AD, Murthy ASV. Perturbed soliton and director deformation in a ferronematic liquid crystal. Chaos Solitons Fractals. 2018;106:220–226. doi: 10.1016/j.chaos.2017.11.022
   Web of Science ®Google Scholar
• Barone A, Esposito F, Magee CJ, et al. Theory and application of the Sine-Gordon equations. Riv Nuovo Cimento. 1971;1(2):227–267. doi: 10.1007/BF02820622
   Google Scholar
• Braun OM, Kivshar YS. Nonlinear dynamics of the Frenkel-Kontorova model. Phys Rep. 1998;306:1–108. doi: 10.1016/S0370-1573(98)00029-5
   Web of Science ®Google Scholar
• Braun OM, Kivshar YS. The Frenkel-Kontorova model: concepts, methods and applications. Berlin: Springer; 2004.
   Google Scholar
• Kivshar YS, Malomed BA. Dynamics of solitons in nearly integrable systems. Rev Mod Phys. 1989;61:763–911. doi: 10.1103/RevModPhys.61.763
   Web of Science ®Google Scholar
• Eisenhart LP. A treaties on the differential geometry of curves and surfaces. Boston (MA): Ginn and Co.; 1909.
   Google Scholar
• Scott RJ. Report of the committee on waves, Report of the seventh meeting of the British association for the advancement of science, Liverpool: 1838. pp. 417–496. Russell J. Scott, Report on Waves, Report of the 14th meeting of the British association for the advancement of science, John Murray, London: 1844. pp. 311–390.
   Google Scholar
• Zabusky NJ, Kruskal MD. Interaction of solitons in a collision-less plasma and the recurrence of initial states. Phys Rev Lett. 1965;15:240–243. doi: 10.1103/PhysRevLett.15.240
   Web of Science ®Google Scholar
• Drazin PG, Johnson RS. Solitons: an introduction. Cambridge University Press; 1989.
   Google Scholar
• Remoissenet M. Waves called solitons: concepts and experiments. Springer; 2013.
   Google Scholar
• Filippov AT. The versatile soliton. Springer; 2010.
   Google Scholar
• Dai C-Q, Yu F-B. Soliton solutions with power-law nonlinearity in inhomogeneous media. Phys Scr. 2013;87:045002. (5pp) doi: 10.1088/0031-8949/87/04/045002
   Web of Science ®Google Scholar
• Degasperis A, Manakov SV, Zenchuk AI. Multidimensional soliton equations in inhomogeneous media. Phys Lett A. 1990;147(1).
   Google Scholar
• Shyu W, Guzdar PN, Chen HH, et al. Transition from soliton to chaos in nonlinear inhomogeneous media. Phys Lett A. 1998;249:307–314. doi: 10.1016/S0375-9601(98)00574-X
   Google Scholar
• Guerrero LE, Bellorin A. Soliton structure dynamics in inhomogeneous media. Physica A. 1998;260:418–424. doi: 10.1016/S0378-4371(98)00331-8
   Google Scholar
• Gonzalez JA, de Mello B. Kink catastrophes. Phys. Scripta. 1996;54:14. doi: 10.1088/0031-8949/54/1/002
   Web of Science ®Google Scholar
• Gharaati A, Khordad R. Dynamics of generalized Sine-Gordon soliton in inhomogeneous media. Indian J Phys. 2011;85(3):433–445. doi: 10.1007/s12648-011-0048-0
   Web of Science ®Google Scholar
• Hasegawa A, Kodama Y. Guiding-center soliton. Phys Rev Lett. 1991;66:161. doi: 10.1103/PhysRevLett.66.161
   PubMed Web of Science ®Google Scholar
• Liu X, Triki H, Zhou Q, et al. Generation and control of multiple solitons under the influence of parameters. Nonlinear Dyn. 2019;95:143. doi: 10.1007/s11071-018-4556-8
   Web of Science ®Google Scholar
• Jagtap AD, Saha E, George JD, et al. Revisiting the inhomogeneously driven Sine-Gordon equation. Wave Motion. 2017;73:76–85. doi: 10.1016/j.wavemoti.2017.05.003
   Web of Science ®Google Scholar
• Argyris J, Haase M, Heinrich JC. Finite element approximation to two-dimensional Sine-Gordon solitons. Comput Methods Appl Mech Eng. 1991;86:1–26. doi: 10.1016/0045-7825(91)90136-T
   Web of Science ®Google Scholar
• Dehghan MGhesmati A. Numerical simulation of two-dimensional Sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Comput Phys Comm. 2010;181:772–786. doi: 10.1016/j.cpc.2009.12.010
   Web of Science ®Google Scholar
• Dehghan M, Mirzaei D. The dual reciprocity boundary element method for two-dimensional Sine-Gordon equation. Comput Methods Appl Mech Engrg. 2008;197:476–486. doi: 10.1016/j.cma.2007.08.016
   Web of Science ®Google Scholar
• Mirzaei D, Dehghan M. Meshless local Petrov-Galerkin (MLPG) approximation to the two-dimensional Sine-Gordon equation. J Comput Appl Math. 2010;233:2737–2754. doi: 10.1016/j.cam.2009.11.022
   Web of Science ®Google Scholar
• Mirzaei D, Dehghan M. Bounday element solution of two-dimensional Sine-Gordon equation using continuous linear elements. Eng Anal Bound Elem. 2009;33:12–24. doi: 10.1016/j.enganabound.2008.03.011
   Web of Science ®Google Scholar
• Bratsos AG. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math Comput Simulation. 2007;76:271–282. doi: 10.1016/j.matcom.2006.11.004
   Web of Science ®Google Scholar
• Sheng Q, Khaliq AQM, Voss DA. Numerical simulation of two-dimensional Sine-Gordon solitons via a split cosine scheme. Appl Math Comput Simul. 2005;68:355–73. doi: 10.1016/j.matcom.2005.02.017
   Web of Science ®Google Scholar
• Rashidinia J, Mohammadi R. Tension spline solution of Sine-Gordon equation. Numer Algorithms. 2011;56:129–42. doi: 10.1007/s11075-010-9377-x
   Web of Science ®Google Scholar
• Cui M. High order compact alternating direction implicit method for the generalized sine-Gordon equation. J Comput Appl Math. 2010;235:837–49. doi: 10.1016/j.cam.2010.07.016
   Web of Science ®Google Scholar
• Mittal RC, Bhatia R. Numerical solution of nonlinear Sine Gordon equation by modified cubic spline collocation method. Int J PDE. 2014;2014:1–8.
   Google Scholar
• Cheng RJ, Liew KM. Analyzing two-dimensional sine Gordon equation with the mesh-free reproducing kernel particle Ritz method. Comput Methods Appl Mech Eng. 2012;245-246:132–143. doi: 10.1016/j.cma.2012.07.010
   Web of Science ®Google Scholar
• Pekmen B, Tezer-Sezgin M. Differential quadrature solution of nonlinear Klein Gordon and Sine-Gordon equations. Comput Phys Commun. 2012;183(8):1702–1713. doi: 10.1016/j.cpc.2012.03.010
   Web of Science ®Google Scholar
• Jiwari R, Pandit S, Mittal RC. Numerical simulation of two-dimensional Sine-Gordon solitons by differential quadrature method. Comput Phys Comm. 2012;183:600–616. doi: 10.1016/j.cpc.2011.12.004
   Web of Science ®Google Scholar
• Li X, Zhang S, Wang Y, et al. Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized Sinh-Gordon equations. Comput Math Appl. 2016;71(8):1655–1678. doi: 10.1016/j.camwa.2016.03.007
   Web of Science ®Google Scholar
• Li X, Li S. Analysis of the complex moving least squares approximations and the associated element-free Galerkin method. Appl Math Model. 2017;47:45–62. doi: 10.1016/j.apm.2017.03.019
   Web of Science ®Google Scholar
• Caputo J-G, Dutykh D. Nonlinear waves in networks: model reduction for Sine-Gordon. Phys Rev E. 2014;90:022912. doi: 10.1103/PhysRevE.90.022912
   Web of Science ®Google Scholar
• Dutykh D, Caputo J-G. Wave dynamics on networks: method and application to the sine-Gordon equation. Appl Numer Math. 2018;131:54–71. doi: 10.1016/j.apnum.2018.03.010
   Web of Science ®Google Scholar
• Jagtap AD, Murthy ASV. Higher order scheme for two-dimensional inhomogeneous Sine-Gordon equation with impulsive forcing. Commun Nonlinear Sci Numer Simulat. 2018;64:178–197. doi: 10.1016/j.cnsns.2018.04.012
   Web of Science ®Google Scholar
• Gordon W. Der comptoneffekt nach der Schrödingerschen theorie. Z Phys. 1926;40:117–133. doi: 10.1007/BF01390840
   Google Scholar
• Klein O. Quantentheorie und fnfdimensionale Relativittstheorie. Z Phys. 1926;37:895–906. doi: 10.1007/BF01397481
   Google Scholar
• Andreev AL. Atomic spectroscopy: introduction to the theory of hyperfine structure. Springer; 2006.
   Google Scholar
• Zwillinger D. Handbook of differential equations. Academic Press; 1997.
   Google Scholar
• Ablowitz MJ, Kaup DJ, Newell AC, et al. Method for solving the Sine-Gordon equation. Phys. Rev. Lett.. Jun 1973;30(25):1262–1264. doi: 10.1103/PhysRevLett.30.1262
   Web of Science ®Google Scholar
• Dziewonski AM, Anderson DL. Preliminary reference Earth model. Phys Earth Planetary Interiors. 1981;25:297–356. doi: 10.1016/0031-9201(81)90046-7
   Web of Science ®Google Scholar
• Boyd JP. Chebyshev and fourier spectral methods. 2nd ed. Dover Publications; 2013.
   Google Scholar
• Guo BY. Spectral methods and their applications. World Scientific; 1998.
   Google Scholar
• Shao W, Wu X. Numerical simulation of nonlinear Klein-Gordon and Sine-Gordon equations using the Chebyshev tau meshless method. Comput Phys Comm. 2014;185:1399–1409. doi: 10.1016/j.cpc.2014.02.002
   Web of Science ®Google Scholar
• Yin F, Tian T, Song J, et al. Spectral methods using Legendre wavelets for nonlinear Klein Sine-Gordon equations. J Comput Appl Maths. 2015;275:321–334. doi: 10.1016/j.cam.2014.07.014
   Web of Science ®Google Scholar
• Liu W, Sun J, Wu B. Space-time spectral method for the two-dimensional generalized Sine-Gordon equation. J Math Anal Appl. 2015;427:787–804. doi: 10.1016/j.jmaa.2015.02.057
   Web of Science ®Google Scholar
• Karniadakis GE, Spencer SJ. Spectral/hp element methods for CFD. Oxford: Oxford University Press; 1999.
   Google Scholar
• Canuto C, Hussaini MY, Quarteroni A, et al. Spectral methods in fluid dynamics, Springer, 1987. (Springer Series in Computational Physics).
   Google Scholar
• Taleei A, Dehghan M. A pseudo-spectral method that uses an overlapping multi-domain technique for the numerical solution of sine-Gordon equation in one and two spatial dimensions. Math Meth Appl Sci. 2014;37:1909–1923. doi: 10.1002/mma.2943
   Web of Science ®Google Scholar
• Lindquist JM, Neta B, Giraldo FX. A spectral element solution of the Klein-Gordon equation with high-order treatment of time and non-reflecting boundary. Wave Motion. 2010;47:289–298. doi: 10.1016/j.wavemoti.2009.11.007
   Web of Science ®Google Scholar
• Kirby RM, Sherwin SJ. Aliasing errors due to quadratic nonlinearities on triangular spectral/hp element discretisations. J Eng Math. 2006;56:273–288. doi: 10.1007/s10665-006-9079-5
   Web of Science ®Google Scholar
• Kirby RM, Karniadakis GE. De-aliasing on non-uniform grids: algorithms and applications. J Comput Phys. 2003;191:249–264. doi: 10.1016/S0021-9991(03)00314-0
   Web of Science ®Google Scholar
• Lomtev I, Kirby RM, Karniadakis GE. A discontinous galerkin method in moving domains. 11. Springer, 2000. (Lecture notes in computational science and engineering).
   Google Scholar
• Nair RD. Diffusion experiments with a global discontinuous galerkin shallow-water model. Mon Weather Rev. 2009;137:3339–3350. doi: 10.1175/2009MWR2843.1
   Web of Science ®Google Scholar
• Richtmyer RD, Morton KW. Difference methods of initial-value problems. Wiley; 1967.
   Google Scholar
• Jagtap AD, Murthy ASV. Higher order spectral element scheme for two- and three-dimensional Cahn-Hilliard equation. Int J Adv Eng Sci Appl Math. 2018;10(1):79–89. doi: 10.1007/s12572-018-0210-4
   Web of Science ®Google Scholar
• Lamb GL. Elements of soliton theory. New York (NY): John Wiley; 1980.
   Google Scholar
• Dodd RK, Morris HC, Eilbeck JC, et al. Solitons and nonlinear wave equations. London: Academic Press; 1982.
   Google Scholar
• Bogolyubskii IL, Makhankov VG. Lifetime of pulsating solitons in certain classical models. JETP Lett. 1976;24(1):12–14.
   Web of Science ®Google Scholar
• Bogolyubskii IL. Oscillating particle like solution of Klein-Gordon equation. JETP Lett. 1976;24(10):535–538.
   Web of Science ®Google Scholar
• Christiansen PL, Olsen OH. Return effect for rotationally symmetric solitary wave solutions to the Sine-Gordon equation. Phys Lett. 1978;68A(2):185–188. doi: 10.1016/0375-9601(78)90798-3
   Google Scholar