Approximate Soliton Solutions of Real Order Sine- Gordon Equations

Year 2017, Volume: 17 Issue: 2, 415 - 425, 31.08.2017

Neslihan Üzar

Abstract

In this study, the fractional Sine-Gordon (SG) equations (time-fractional, space-fractional and timespace-

fractional) are solved using Homotopy Perturbation Method (HPM). The crucial point is the

attained remarkable result from these solutions. While the solutions of classical and time-fractional SG

equations are kink of type (Although of being the same type, they are different from each other),

solution of the space-fractional SG equation is breather of type i.e., different types of soliton solutions

are obtained using similar initial conditions for time and space fractional SG equation. Also these results

show that some events such as vortex-antivortex couples in a Josephson junction or losses in signal

dispersion of fiber optics communication can be modelled by fractional SG equations. In other words,

this study may be very important for bringing to light the real behaviour of physical systems which have

usually been described by classical SG equation. Because some physical events such as the memory

effects of non-Markovian processes, the effects of non-Gaussian distribution, interactions between the

systems and environment and some physical losses in the systems which are neglected in classical SG

equation can be taken into account with fractional SG equations.

Keywords

Caputo fractional derivative operator, HPM, SG equation, fractional non-linear equation

References

• Akgul A., Inc M., Karatas E. and Baleanu D., 2015. Numerical solutions of fractional differential Approximate Soliton Solutions of Real Order Sine- Gordon Equations, Üzar 424 equations of Lane-Emden type by an accurate technique.Advances in Difference Equations, 2015, 220.
• Akgul A., Inc M., Kilicman A. and Baleanu D., 2016. A new approach for one-dimensional sine-Gordon equation. Advances in Difference Equations, 2016, 8. Aktosun T., Demontis F. and Mee C., 2010. Exact solutions to the Sine-Gordon Equation. Journal of Mathematical Physics, 51, 123521.
• Ateş I. and Yıldırım A., 2010. Applications of variational iteration and homotopy perturbation methods to obtain exact solutions for time-fractional diffusion-wave equations. International Journal of Numerical Methods for Heat & Fluid Flow, 20(6), 638.
• Batiha B. et al., 2007. Numerical solution of sine-Gordon equation by variational iteration method. Physics Letters A, 370, 437-440.
• Bracken P., 2011. Surfaces of Arbitrary Constant Negative Gaussian Curvature and Related Sine-Gordon Equations. Mathematica Aeterna, 1 (11), 1-11. Caputo M.,1967. Linear model of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc., 13, 529-539.
• Derks G., Doelman A., Gils S., Visser T., 2003 Travelling waves in a singularly perturbed sine-Gordon equation. Physica D, 180, 40-70.
• Elsaid A., Hammad D., 2012. Measurable Lipschitz selections and set-valued integral equations of fractional order. Journal of Fractional Calculus and Applications, 2, 1-8.
• El-Shahed M., 2005. Application of He's homotopy–perturbation method to Volterra's integro-differential equation. Int. J. Nonlinear Sci. Numer. Simul., 6, 163-168.
• Frenkel J., Kontorova T., 1939. On the theory of plastic deformation and twinning. Journal of Physics (USSR), 1, 137-149.
• He J. H., 2004. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. Math. Comput, 151, 287-292.
• He J. H., 2000. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Internat. J. Non-Linear Mech., 35, 37-43.
• He J. H., 2005. Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals, 26, 695-700.
• He J. H., 2005, Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul, 6, 207-208.
• He J. H., 2005. Periodic solutions and bifurcations of delay differential equations. Phys. Lett. A, 374, 228-230.
• Ivancevic Vladimir G and Ivancevic Tijana T., 2013. Sine-Gordon Solitons, Kinks and Breathers as Physical Models of Nonlinear Excitations in Living Cellular Structures. Journal of Geometry and Symmetry in Physics (JGSP), 31, 1-56.
• Jin L. 2009. Analytical approach to the sine-Gordon equation using homotopy perturbation method. Int. J. Contemp. Math. Sciences, 4 (5) 225-231.
• Kaya D 2003. A numerical solution of the sine-Gordon equation using the modified decomposition method. Applied Mathematics and Computation, 143, 309 .
• Lu J., 2009.An analytical approach to the sine–Gordon equation using the modified homotopy perturbation method. Computers and Mathematics with Applications, 58, 2313–2319.
• Metzler R., Klafter J., 2000. The random walks guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep., 339, 1-77.
• Miller K. S, Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, John Wiley Sons. Miskinis L., 2005. The Nonlinear and Nonlocal Integrable Sine-Gordon Equation. Mathematical Modelling and Analysis, 479, 483.
• Momani S., Odibat Z., 2007. Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A, 365, 345.
• Nasrolahpour H., 2013. Fractional Lagrangian and Hamiltonian formulations in field theory Generalized multiparameters fractional variational calculus. Prespacetime Journal, 4, 604-608.
• Oldham K. B, Spainer J., 1974. The Fractional Calculus. San Diego California, Academic Press.
• Pandir Y.and Duzgun H. H., 2016. New exact solutions for fractional Sine-Gordon equation by using the new version of generalized F-expansion method. AIP Conference Proceedings, 1738, 290020.
• Podlubny I., 1999. Fractional Differential Equations. New York, Academic Press.
• Saha Ray S., 2016. A new analytical modelling for nonlocal generalized Riesz fractional sine-Gordon equation. Journal of King Saud University – Science, 28(1), 48-54.
• Tarasov V.E., 2005. Continuous medium model for fractal media. Phys. Lett. A, 336, 167 . Tsallis C., 1988. Possible generalization of Boltzmann-Gibbs statistics. Stat. Phys, 52, 479–487.
• Yousef A. M. , Rida S. Z. and Ibrahim H. R., 2016. Solutions for the fractional differential coupled sine- Gordon equation with Homotopy analysis method and the modified decomposition method. Scitech Research Organisation, 6(4), 831.
• Wazwaz A. M., 2012. N-soliton solutions for the sine-Gordon equation of different dimensions. J. Appl. Math. & Informatics, 30 (5-6), 925-934.
• 1-http://pauli.uni-muenster.de/tp/fileadmin/lehre/NumMethoden/WS0910/ScriptPDE, (10.06.2015)
• 2-http://www.fas.org/sgp/othergov/doe/lanl/pubs/00285753.pdf, (26.02.2016)
• 3-http://young.physics.ucsc.edu/250/mathematica/sinegordon.nb.pdf, (03.05.2014)
• 4- http://www.researchgate.net/publication/241918156, (28.04.2016)