The Modified (G'/G)-Expansion Method for Exact Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation

Year 2012, Volume: 9 Issue: 1, - , 01.02.2012

Reza Abazari

Abstract

In this paper, we successfully modified the (G0/G)-expansion method and
as an application proposed to construct exact solutions of the (3+1)-dimensional JimboMiwa
equation. Each of the obtained solutions, namely the hyperbolic function solutions,
the trigonometric function solutions and the rational solutions contain an explicit linear
function of the variables in the equation in question. It is shown that the proposed method
with the help of a symbolic computation provides a more powerful mathematical tool for
solving nonlinear evolution equations in mathematical physics.

Keywords

Nonlinear evolution equation, Jimbo-Miwa equation, (G0/G)-expansion method, hyperbolic function solutions, trigonometric function solutions, rational solutions

References

• [1] G. T. Liu and T. Y. Fan, New applications of developed Jacobi elliptic function expansion methods, Physics Letters A 345 (2005), 161–166.
• [2] M. J. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform, SIAM, Philadelphia 1981.
• [3] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge 2004.
• [4] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Physics Letters A 213 (1996), 279–287.
• [5] J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation 151 (2004), 287–292.
• [6] Z. Y. Yan, An improved algebra method and its applications in nonlinear wave equations, Chaos, Solitons & Fractals 21 (2004), 1013–1021.
• [7] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York 1989.
• [8] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston 1994.
• [9] R. Abazari and M. Abazari, Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM, Communications in Nonlinear Science and Numerical Simulation 17 (2012), 619–629.
• [10] H. Jafari, A. Borhanifar and S. A. Karimi, New solitary wave solutions for the bad Boussinesq and good Boussinesq equations, Numerical Methods for Partial Differential Equations 25 (2009), 1231–1237.
• [11] A. Borhanifar, M. M. Kabir and L. Maryam Vahdat, New periodic and soliton wave solutions for the generalized Zakharov system and (2+ 1)-dimensional Nizhnik-Novikov-Veselov system, Chaos, Solitons & Fractals 42 (2009), 1646–1654.
• [12] A. Borhanifar and M. M. Kabir, New periodic and soliton solutions by application of Expfunction method for nonlinear evolution equations, Journal of Computational and Applied Mathematics 229 (2009), 158–167.
• [13] T. Ozi¸s and ¨ ˙I. Aslan, Exact and explicit solutions to the (3 + 1)-dimensional Jimbo-Miwa equation via the Exp-function method, Physics Letters A 372 (2008), 7011–7015.
• [14] M. Wang, X. Li and J. Zhang, The ( G'/G)–expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A 372 (2008), 417–423.
• [15] A. Bekir, Application of the ( G'/G )–expansion method for nonlinear evolution equations, Physics Letters A 372 (2008), 3400–3406.
• [16] ˙I. E. ˙Inan, ( G'/G)-Expansion method for traveling wave solutions of the sixth-order Ramani equation, C¸ ankaya University Journal of Science and Engineering 7 (2010), 51–57.
• [17] R. Abazari, The ( G'/G )-expansion method for the coupled Boussinesq equation, Procedia Engineering 10 (2011), 2845–2850.
• [18] R. Abazari, Application of ( G'/G )–expansion method to travelling wave solutions of three nonlinear evolution equation, Computers & Fluids 39 (2010), 1957–1963.
• [19] R. Abazari, The ( G'/G )–expansion method for Tzitz´eica type nonlinear evolution equations, Mathematical and Computer Modelling 52 (2010), 1834–1845.
• [20] R. Abazari and R. Abazari, Hyperbolic, trigonometric and rational function solutions of Hirota–Ramani equation via ( G'/G )-expansion method, Mathematical Problems in Engineering 2011 (2011), Article ID 424801, doi:10.1155/2011/424801. http://downloads.hindawi.com/journals/mpe/2011/424801.pdf, 2010. Online; accessed 05-May-2012.
• [21] R. Abazari, General travelling wave solutions of quintic nonlinearity of Klein-Gordon equation, Australian Journal of Basic and Applied Sciences 5 (2011), 197–205.
• [22] R. Abazari, The solitary wave solutions of Zoomeron equation, Applied Mathematical Sciences 5 (2011), 2943–2949.
• [23] M. M. Kabir, A. Borhanifar and R. Abazari, Application of ( G'/G )–expansion method to Regularized Long Wave (RLW) equation, Computers & Mathematics with Applications 61 (2011), 2044–2047.
• [24] C. P. Liu, The relation between the kink-type solution and the kink-bell-type solution of nonlinear evolution equations, Physics Letters A 312 (2003), 41–48.