Symmetry Reductions, Exact Solutions and Conservation Laws for the Coupled Nonlinear Klein-Gordon System

Year 2017, Volume: 13 Issue: 3, 593 - 599, 30.09.2017

Emrullah Yaşar İlker Burak Giresunlu

https://izlik.org/JA33UD76RY

Abstract

The Lie group method is applied to a coupled nonlinear Klein-Gordon
system. The Klein-Gordon system is used to model many nonlinear phenomena
including the propagation of dislocations in crystals and the behavior of
elementary particles and the propagation of fluxons in Josephson junctions. The
symmetry reductions and exact solutions which include the stationary and
solitary waves are obtained. In addition, by using the multiplier method, we
derive the local conservation laws of the coupled nonlinear Klein-Gordon
system.

Keywords

Lie symmetry analysis , Coupled nonlinear Klein-Gordon system , Similarity reduction , Solitary wave solution

References

• Vakhnenko, V.O, Parkes, E.J, Morrison, A.J, A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos, Soliton and Fractals. 2003, 17, 683-92.
• Hirota, R, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Physical Review Letters. 1971, 27, 1192-1195.
• Ma, Y, Geng, X, Darboux and Bäcklund transformations of the bidirectional Sawada--Kotera equation. Applied Mathematics and Computation. 2012, 218, 6963-6967.
• Zerarka, A, Ouamane, S, Attaf, S, On the functional variable method for finding exact solutions to a class of wave equations, Applied Mathematics and Computation. 2010, 217, 2897.
• Wang, M.L, Zhou, Y.B, Li, Z.B, Application of a homogeneous balance method to exact solutions of nonlinear equation in mathematical physics. Physics Letters. A. 1996, 216, 67-75.
• Bekir, A, Applications of the extended tanh method for coupled nonlinear evolution equations. Communications in Nonlinear Science and Numerical Simulation. 2008, 13, 1748-1757.
• Bekir, A, Application of the (G'⁄G)-expansion method for nonlinear evolution equations. Physics Letters A. 2008, 372, 3400-3406.
• Olver, P.J, Application of Lie groups to Differential Equations. Springer-Verlag, New York. 1993.
• Tascan, F, Bekir A, Koparan, M, Travelling wave solutions of nonlinear evolution equations by using the first integral method, Communications in Nonlinear Science & Numerical Simulation. 2009, 14, 1810-1815.
• Moore, K.R, Coupled Boussinesq equations and nonlinear waves in layered waveguides. (Doctoral dissertation, © KR Moore), 2013.
• Khusnutdinova, K.R, Coupled Klein-Gordon equations and energy exchange in two-component systems. The European Physical Journal Special Topics. 2007, 147(1), 45-72.
• Wazwaz, A.M, New travelling wave solutions to the Boussinesq and the Klein-Gordon equations, Communications in Nonlinear Science and Numerical Simulation. 2008, 13, 889-901.
• Chen, J, Yang, J, Yang, H, Single soliton solutions of the coupled nonlinear Klein-Gordon equations with power law nonlinearity. Applied Mathematics and Computation. 2014, 246, 184-191.
• Hirota, R, Ohta, Y, Hierarchies of coupled soliton equations. I, Journal of the Physical Society of Japan. 1991, 60, 798-809.
• Alagesan, T, Chung, Y, Nakkeeran, K, Soliton solutions of coupled nonlinear Klein-Gordon equations. Chaos, Solitons & Fractals. 2004, 21, 879-882.
• Alagesan, T, Chung, Y, Nakkeeran, K, Painlevé analysis of N-coupled nonlinear Klein-Gordon equations. Journal of the Physical Society of Japan. 2003, 72, 1818.
• Yusufoğlu, E, Bekir, A, Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine-cosine method, International Journal of Computer Mathematics. 2006, 83, 915-924.
• Zhang, Z, New Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation. Turkish Journal of Physics. 2008, 32, 235-240.
• Porsezian, K, Alagesan, T, Painlevé analysis and complete integrability of coupled Klein-Gordon equations. Physics Letters A. 1995, 198, 378-382.
• Duncan, D.B, Symplectic finite difference approximations of the nonlinear Klein-Gordon equation, SIAM Journal of Numerical Analysis. 1997, 34, 1742-1760.
• Wazwaz, A.M, The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation, Applied Mathematics and Computation. 2005, 167, 1179-1195.
• Anderson, I.M, Fels, M.E, Torre, C.G, Group invariant solutions without transversality, Communications in Mathematical Physics. 2000, 212, 653-686.
• Bluman, G.W, Anco, S.C, Symmetry and Integration Methods for Differential Equations. vol. 154 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2002.
• Stephani, H, Differential Equations and their Solutions using symmetries (M. MacCallum, ed.), Cambridge University Press, Cambridge, 1989.
• Vorob'ev, E.M, Reduction of quotient equations for differential equations with symmetries, Acta Applicandae Mathematicae. 1991, 23.
• Winternitz, P, Group theory and exact solutions of partially integrable equations, Partially Integrable Evolution Equations (R. Conte and N. Boccara, eds.), Kluwer Academic Publishers, 1990.
• Ibragimov, N.H, CRC Handbook of Lie Group Analysis of Differential Equations, Volume 1, Symmetries, Exact Solutions and Conservation Laws., CRC Press, Boca Raton, Florida, 1995.
• Rogers, C, Shadwick, W, Nonlinear boundary value problems in science and engineering, Mathematics in Science and Enginering, vol. 183 (W. F. Ames, eds.), Academic Press, Boston, 1989.
• Ibragimov, N.H, Khamitova, R, Thide, B, Conservation laws for the Maxwell-Dirac equations with dual Ohm's law, Journal of Mathematical Physics. 2007, 48, 053523.
• Cheviakov, A.F, Gem software package for computation of symmetries and conservation laws of differential equations. Computer Physics Communications. 2007, 176, 48-61.
• Steudel, H, Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen. Zeitschrift für Naturforschung. 1962, 17A, 129-132.
• Kara, A.H, Mahomed, F.M, Relationship between symmetries and conservation laws, International Journal of Theoretical Physics. 2000, 39, 23-40.
• Kara, A.H, Mahomed, F.M, Noether-type symmetries and conservation laws via partial Lagragians, Nonlinear Dynamics. 2006, 45, 367-383.
• Anco, S.C, Bluman, G.W, Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications, European Journal of Applied Mathematics. 2002, 13, 545-566.
• Adem, A.R, Khalique, C.M, New exact solutions and conservation laws of a coupled Kadomtsev-Petviashvili system, Computers & Fluids. 2013, 81, 10-16.
• Adem, A.R, Khalique, C.M, Symmetry reductions, exact solutions and conservation laws of a new coupled KdV system, Communications in Nonlinear Science & Numerical Simulation. 2012, 17, 3465-3475.
• Johnpillai, A.G, Khalique, C.M, Mahomed, F.M, Travelling wave group-invariant solutions and conservation laws for θ-equation, Symmetries, Differential Equations and Applications II (SDEA II) 2014 Islamabad, Pakistan.
• Ibragimov, N.H, A new conservation theorem, Journal of Mathematical Analysis and Application. 2007, 333, 311-328.
• Magalakwe, G, Muatjetjaja, B, Khalique, C.M, Generalized double sinh-Gordon equation: Symmetry reductions, exact solutions and conservation laws. Iranian Journal of Science and Technology (Sciences), 2015, 39(3), 289-296.
• Kudryashov, N. A, Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Soliton and Fractals. 2005, 24, 1217-31.
• Kudryashov, N. A, Exact solitary waves of the Fisher equation. Physics Letters A. 2005, 342, 99-106.
• Jafari, N, Kadkhoda, N, Kahlique, C.M, Exact Solutions of d^4 Equation Using Lie Symmetry Approach along with the Simplest Equation and Exp-Function Methods, Abstract and Applied Analysis. 2012, 350287.
• LeVeque, R.J, Numerical methods for Conservation laws. Lec. in Math., 1992.
• Wolf, T, A comparison of four approaches to the calculation of conservation laws, European Journal of Applied Mathematics, 2002, 02, 129-152.