Exact wave solutions to the (2+1)-dimensional Klein-Gordon equation with special types of nonlinearity
Authors:Sk. Tanzer Ahmed Siddique,Md. Dulal Hossain,M. Ali Akbar,
In this article, we investigate the traveling wave solutions to the Klein-Gordon equation in (2+1)-dimension with special types of nonlinearity. The types include quadratic, cubic and polynomial nonlinearity. The Klein-Gordon equation assumes significant role in numerous types of scientific investigation such as in quantum field theory, nonlinear optics, nuclear physics, magnetic field etc. To investigate the aimed traveling wave solutions, we execute the (?′/?)-expansion method. The attained solutions are in the form of hyperbolic, trigonometric and rational functions. The results acknowledged that the applied method is very efficient and suitable for discovering differential equations with various types of nonlinearity considered in optics and quantum field theory. The solutions of the Klein-Gordon equation with quadratic, cubic, and polynomials nonlinearity play a significant role in many scientific measures notably optics and quantum field theory.
Keywords:Klein-Gordon equation,nonlinearity,travelingwave solutions,
I.Abdou, M., The extended tanh method and its applications for solving nonlinear physical models.Applied Mathematics and Computation, 2007. 190(1): p. 988-996.
II.Ablowitz, M.J., et al., Solitons, nonlinear evolution equations and inverse scattering. Vol. 149. 1991: Cambridge university press.
III.Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method Kluwer.Boston, MA, 1994.
IV.Akbar, M.A., N. Hj, and M. Ali. New solitary and periodic solutions of nonlinear evolution equation by Exp-function method. in World Appl. Sci. J. 2012. Citeseer.
V.Akter, J. and M.A. Akbar, Exactsolutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method.Results in physics, 2015. 5: p. 125-130.
VI.Alfimov, G., et al., Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential.Physical Review E, 2002. 66(4): p. 046608.
VII.Biswas, A., C. Zony, and E. Zerrad, Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation.Applied Mathematics and Computation, 2008. 203(1): p. 153-156.
VIII.Biswas, A., et al., Solitons and conservation laws of Klein–Gordon equation with power law and log law nonlinearities.Nonlinear Dynamics, 2013. 73(4): p. 2191-2196.
IX.Cataliotti, F., et al., Superfluid current disruption in a chain of weakly coupled Bose-Einstein condensates.New Journal of Physics, 2003. 5(1): p. 71.
X.El-Wakil, S., M. Abdou, and A. Hendi, New periodic wave solutions via Exp-function method.Physics Letters A, 2008. 372(6): p. 830-840.
XI.Guy, J., Lagrange characteristic method forsolving a class of nonlinear partial differential equations of fractional order.Applied Mathematics Letters, 2006. 19(9): p. 873-880.
XII.Khan, K. and M.A. Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method.Ain Shams Engineering Journal, 2013. 4(4): p. 903-909.
XIII.Kumar, A., S. Kumar, and M. Singh, Residual power series method for fractional Sharma-Tasso-Olever equation.Communications in Numerical Analysis, 2016. 2016(1): p. 1-10.
XIV.Mohyud-Din, S.T., A. Yildirim, and S.A. Sezer, Numerical soliton solutions of improved Boussinesq equation.International Journal of Numerical Methods for Heat & Fluid Flow, 2011. 21(7): p. 822-827.
XV.Naher, H., F.A. Abdullah, and M.A. Akbar, New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method.Journal of Applied Mathematics, 2012. 2012.
XVI.Sassaman, R. and A. Biswas,Topological and non-topological solitons of the Klein-Gordon equations in 1+ 2 dimensions.Nonlinear Dynamics, 2010. 61(1-2): p. 23-28.
XVII.Shakeri, F. and M. Dehghan, Numerical solution of the Klein–Gordon equation via He’s variational iteration method.Nonlinear Dynamics, 2008. 51(1-2): p. 89-97.
XVIII.Sirendaoreji, Exact travelling wave solutions for four forms of nonlinear Klein-Gordon equations.PHYSICS LETTERS A, 2007. 363(5-6): p. 440-447.
XIX.Wazwaz, A.-M., New travelling wave solutionsto the Boussinesq and the Klein–Gordon equations.Communications in Nonlinear Science and Numerical Simulation, 2008. 13(5): p. 889-901.
XX.Wazwaz, A.-M., The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integrodifferential equations.Applied Mathematics and Computation, 2010. 216(4): p. 1304-1309.
XXI.Yan, C., A simple transformation for nonlinear waves.Physics Letters A, 1996. 224(1-2): p. 77-84.
XXII.Yasuk, F., A. Durmus, and I. Boztosun, Exact analytical solution to the relativistic Klein-Gordon equation with noncentral equal scalar and vector potentials.Journal of mathematical physics, 2006. 47(8): p. 082302.
XXIII.Zheng, Y. and S. Lai, A study on three types of nonlinear Klein-Gordon equations.Dynamics of Continuous, Discrete and Impulsive Systems: Series B, 2009. 16(2): p. 271-279.View Download