Exact travelling wave solutions for some complex nonlinear partial differential equations

Document Type : Research Paper

Authors

1 University of Guilan

2 Department of Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

3 University of Mazandaran

Abstract

This paper reflects the implementation of a reliable technique which is called $left(frac{G'}{G}right)$-expansion  ethod for  constructing exact travelling wave solutions of nonlinear partial  differential equations. The proposed algorithm has been successfully tested on two two selected equations, the balance numbers of which are not positive integers namely Kundu-Eckhaus equation and  Derivative nonlinear Schr"{o}dinger’s equation. This method is a powerful tool for searching exact travelling solutions in closed form.

Keywords

References

[1] H. Naher, F. A. Abdullah, The improved ( G′/G)-expansion method to the (2+1)-dimensional breaking soliton equation. Journal of Computational Analysis & Applications, 16(2), (2014) 220-235.
[2] H. Naher, F. A. Abdullah, New generalized and improved ( G′/G)-expansion method for nonlinear evolution equations in mathematical physics. Journal of the Egyptian Mathematical Society, http://dx.doi.org/10.1016/j.joems.2013.11.008.
[3] H. Naher, F. A. Abdullah, New approach of ( G′/G)-expansion method and new approach of generalized ( G′/G)-expansion method for nonlinear evolution equation, AIP Advances, 3(3) (2013) 032116.
[4] M.L. Wang, X.Z. Li, J.L. Zhang, The ( G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A, 372 (2008) 417-423.
[5] W.M. Taha, M.S.M. Noorani, I. Hashim, New exact solutions of sixth-order thin-film equation.
Journal of King Saud University- Science, 26 (2014) 75-78.
[6] G. Ebadi, A. Biswas, The ( G′/G) method and topological soliton solution of the K(m, n) equation. Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2377-2382.
[7] E. Zayed, K.A. Gepreel, Some applications of the ( G′/G)-expansionmethod to non-linear partial differential equations. Appl. Math. Comput. 212(1) (2009) 113.
[8] M. Mirzazadeh, M. Eslami, A. Biswas, Soliton solutions of the generalized Klein-Gordon equation by using ( G′/G)-expansion method, Comp. Appl. Math. DOI 10.1007/s40314-013-0098-3.
[9] M. Eslami, M. Mirzazadeh, A. Biswas, Soliton solutions of the resonant nonlinear Schrodinger’s equation in optical fibers with time-dependent coefficients by simplest equation approach.Journal of Modern Optics, 60(19) (2013) 1627-1636.
[10] N. Taghizadeh, M. Mirzazadeh, The simplest equation method to study perturbed nonlinear Schrodingers equation with Kerr law nonlinearity. Commun. Nonlinear Sci. Numer Simulat. 17 (2012) 1493-1499.
[11] A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaei, A. Biswas, New exact travelling wave solutions for DS-I and DS-II equations. Nonlinear Anal.: Modell. Control, 17 (3) (2012) 369-378.
[12] A. Biswas, Optical Solitons with Time-Dependent Dispersion, Nonlinearity and Attenuation in a Kerr-Law Media. Int. J. Theor. Phys. 48 (2009) 256-260.
[13] A. Biswas, 1-Soliton solution of the K(m,n) equation with generalized evolution. Phys. Lett. A. 372(25) (2008) 4601-460.
[14] A. Biswas, 1-Soliton solution of the K(m,n) equation with generalized evolution and timedependent damping and dispersion. Comput. Math. Appl. 59(8) (2010) 2538-2542.
[15] M. Eslami, M. Mirzazadeh, Topological 1-soliton solution of nonlinear Schrodinger equation with dual-power law nonlinearity in nonlinear optical fibers. Eur. Phys. J. Plus, (2013) 128-140.
[16] N. Taghizadeh, M. Mirzazadeh, A. Samiei Paghaleh, Exact solutions of some nonlinear evolution equations via the first integral method. Ain Shams Engineering Journal, 4 (2013) 493-499.
[17] F. Tascan, A. Bekir, M. Koparan, Travelling wave solutions of nonlinear evolutions by using the first integral method. Commun. Nonlinear Sci. Numer Simul. 14 (2009) 1810-1815.
[18] F. Tascan, A. Bekir, Travelling wave solutions of the Cahn-Allen equation by using first integral method. Appl. Math. Comput. 207 (2009) 279-282.
[19] A. Nazarzadeh, M. Eslami and M. Mirzazadeh, Exact solutions of some nonlinear partial differential equations using functional variable method, Pramana J. Phys. 81 (2013) 225-236.
[20] M. Mirzazadeh, M. Eslami, Exact solutions for nonlinear variants of Kadomtsev-Petviashvili (n,n) equation using functional variable method.Pramana J. Phys. 81 (2013) 225-236.
[21] A.C. Cevikel, A. Bekir, M. Akar and S. San, A procedure to construct exact solutions of nonlinear evolution equations. Pramana J. Phys. 79(3) (2012) 337-344.
[22] Dmitry Levko, Alexander Volkov, Modeling of Kundu-Eckhaus equation, (2006), ArXiv:nlin.PS/0702050.
[23] A. Biswas, K. Porsezian, Soliton perturbation theory for the modified nonlinear Schrodingers equation. Commun. Nonlinear Sci. Numer. Simulat. 12 (2007) 886-903.
Computational Methods for Differential Equations
Volume 2, Issue 1 - Serial Number 1
January 2014
Pages 11-18

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History

  • Receive Date: 06 January 2014
  • Revise Date: 23 April 2014
  • Accept Date: 23 April 2014

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