A Modification of the Generalized Kudryashov Method for the System of Some Nonlinear Evolution Equations


H. M. Shahadat Ali,M. A.Habib,M. Mamun Miah,M. Ali Akbar,




Thegeneralized Kudryashov method,Coupled Higgs field equation,Benney-Luke equation,DSW equation, Traveling wave solution,Solitary wave solution,Exact solution,


In this study, a comparatively new technique named the generalized Kudryashov method (gKM) has been effectively implemented to explore the exact traveling wave solutions to some nonlinear evolution equations (NLEEs) in the field of nonlinear science and engineering. The effectiveness of the new functional method has been demonstrated by investigating single as well as coupled equations with arbitrary parameters explicitly the coupled Higgs field equation, the Benney-Luke equation, and the Drinfel'd-Sokolov-Wilson (DSW) equation. As a matter of fact, the solution attained in this article thrust into the abundant wave solutions which includes kink, singular kink, periodic and solitary wave solutions. Moreover, the characteristics of these analytic solutions are interpreted depicting some 2D and 3D graph by using computer symbolic programming Wolfram Mathematica. The computational work ascertained that the employed method is sturdy, simple, precise, and wider applicable. Also, the prominent competence of this current method ensures that practically capable to reducing the size of the computational task and can be solved several nonlinear types of new complex higher order partial differential equations that originating in applied mathematics, computational physics and engineering.


I.A. Bekir,A. Boz, “Applications of the He’s exp function method for nonlinear evolution equations”, Comput. Math. Appl., Vol.: 58, Issue: 11-12, pp.: 2286-2293, 2009.

II.A.H. Arnous, M. Mirzazadeh, M. Eslami,”Exact solution of the Drinfel’d Sokolov Wilson equation using Backlund transformation of Riccati equation and trial function approach”,Prama. J. Phys. Vol.: 86, Issue: 6,pp.: 1153-1160, 2016.

III.A. H. Arnous, M. Mirzazadeh, M. Eslami,”The Backlund transformation method of Riccati equation applied to Coupled Higgs field and Hamiltonian amplitude equations”,Comput. Methods Diff. Equat.,Vol.: 2, Issue: 4,pp.: 216-226, 2014.

IV.A. J. M. Jawad, M. D. Petkovic, A. Biswas,”Modified simple equation method for nonlinear evolution equations”, Appl. Math. Comput., Vol.: 217, Issue: 2,pp.: 869-877, 2010.

V.A. M. Wazwaz, “The extended tanh method for abundant solitary wave solutions of nonlinear wave equations”,Appl. Math. Comput., Vol.:187, Issue: 2,pp.: 1131-1142, 2007.

VI.D. Kumar, A. R. Seadawy, A. K. Joardar,”Modified Kudryashov method via new exact solution for some conformable fractional differential equations arising in mathematical biology”,Chin. J. Phys.,Vol.: 56, Isssue:1,pp.:75-85, 2018.

VII.D. Lu, D. Kang, B. Hong,”New exact solutions of the Drinfel’d SokolovWilson equation”,J. Informa. Comput. Sci., Vol.:18, pp.: 5955-5962, 2013.

VIII.E. Aksoy, M. Kaplan, A. Bekir,”Exponential rational function method for space-time fractional differential equations”,Waves Rand. Compl. Media, Vol.: 26, pp.: 142-151, 2016.

IX.E. Babolian, A. Azizi, J. saeidian,”Some notes on using the homotopy perturbation method for solving time-dependent differential equations”, Math. Comput. Model., Vol.; 50, Issue: 1-2, pp.: 213-224, 2009.

X.E. Fan,”Extended tanh method and its applications to nonlinear equation”. Phys. Lett. A, Vol.: 277, Issue: 4-5,pp.: 212-218, 2000.

XI.E. Fan, J. Zhang,”Applications of the Jacobi elliptic function method to special-type nonlinear equations”,Phys. Lett. A, Vol.:305, Issue: 6, pp.: 383-392, 2002

XII.E. M. E. Zayed,A. G. A. Nowehy,”The solitary wave ansatz method for finding the exact bright and dark soliton solutions of two nonlinear Schrodinger equations”,J. Assn. Arab Univ. Basic Appl. Sci., Vol.: 24, Issue:1, pp.:184-190, 2017.

XIII.F. Mahmud,M. Samsuzzoha, M. A. Akbar, “The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the Fishers equation”,Res. Phys., Vol.: 7, pp.: 4296-4302, 2017.

XIV.G. Allah,R. Musa, Elzaki, M. Tarig, “Application of new homotopy perturbation method for solving partial differential equations”, J. Comput. Theor. Nanosci., Vol.: 15, Issue: 2, pp.: 500-508, 2018.

XV.H. Mao,Q. P. Liu, “Backlund-Darboux transformation and discretizations of 𝑁=2, 𝑎=−2supersymmetric KdV equation”,Phys. Lett. A, Vol.:382, Issue: 5, pp.: 253-258, 2018.

XVI.H. Naher,F. A. Abdullah, M. A. Akbar, “The exp function method for the new exact solution of the nonlinear partial differential equations”, Int. J. Phys. Sci., Vol.: 6, Issue: 29,pp.:6706-6716, 2011.

XVII.H. Triki,A. Yildirim, T. Hayat, O. M. Aldossary, A. Biswas, “Shockwave solution of Benney-Luke equation”,Romanian J. Phys., Vol.: 57, Issue: 7-8, pp.: 1029-1034, 2012.

XVIII.I. Hasim,”Adomian decomposition method for solving BVPs for fourth-order integrodifferential equations”,J. Comput. Appl. Math., Vol.: 193, Issue: 2,pp.:658-664, 2006.

XIX.J. H.He, “Homotopy perturbation technique”,Comput. Methods Appl. Mech. Eng., Vol.:178, Issue: 3-4,pp.:257-262, 1999.

XX.K. A. Gepreel, T. A. Nofal, A. A. Alasmari,”Exact solutions for nonlinear integro-partial differential equations using the generalized Kudryashov method”,J. Egypt. Math. Soc., Vol.: 25, pp.: 438-444, 2017.

XXI.K. Khan, M. A. Akbar, N. H. M. Ali,”The modified simple equation for exact and solitary wave solution of nonlinear evolution equation: the GZK-BBM equation and right-handed non-commutative Burgers equations”,ISRN Math. Phys., pp: 5, Article ID 146704, 2013.

XXII.K. R. Raslan,”The application of He’s exp function method for mKdV and Burgers equations with variable coefficients”,Int. J. Nonlinear Sci., Vol.: 7, Issue: 2, pp.: 174-181, 2009.

XXIII.L. Xu,”He’s parameter expanding methods for strongly nonlinear oscillators”,J. Comput. Appl. Math., Vol.: 207, Issue: 1, pp.: 148-154, 2007.

XXIV.M. A. Akbar, N. H. M. Ali,”The improved F-expansion method with the Riccati equation and its applications in mathematical physics”,Cogent Math. Vol.: 4, ID.: 1282577, 2017

XXV.M. A. Khater, A. R. Seadawy, D. Lu,”Dispersive solitary wave solutions of new coupled Konno-Ono,Higgs field and Maccari equations and their applications”,J. King Saud Univ. Sci., Vol.: 30, pp.: 417-423, 2018.

XXVI.M. Kaplan, A. Bekir, A. Akbulut, E. Aksoy,”The modified simple equation method for nonlinear fractional differential equations”,Romanian J. Phys., Vol.: 60, Issue: 9-10,pp.:1374-1383, 2015.

XXVII.M. K. Elboree,”The Jacobi elliptic function method and its application for two-component BKP hierarchy equations”,Comput. Math. Appl., Vol.: 62, Issue: 12,pp.: 4402-4414, 2011.

XXVIII.M. Koparan, M. Kaplan, A. Bekir, O. Guner,”A novel generalized Kudryashov method for exact solutions of nonlinear evolution equations”,AIP Con. Proc., Vol.: 1798, Issue: 1, 2017.

XXIX.M. M. Kabir, A. Khajeh, E. Aghdam, A. Y. Koma,”Modified Kudryashov method for finding exact solitarywave solutions of higher order nonlinear equations”,Math. Methods Appl. Sci., Vol.: 34, Issue: 2, pp.: 213-219, 2011.

XXX.M. S. Islam, K. Khan, M. A. Akbar,”Application of the improved F-expansion method with Riccati equation to find the exact solution of the nonlinear evolution equations”,J. Egypt. Math. Soc.,Vol.:25, pp.: 13-18, 2017.

XXXI.N. Ahmed, S. Bibi, U. Khan, S. T. Mohyud-din,”A new modification in the exponential rational function method for nonlinear fractional differential equations”,Eur. Phy. J. Plus, Vol.: 133, Issue: 45, 2018.

XXXII.N. Taghizadeh, M. Mirzazadeh,”The first integral method to some complex nonlinear partial differential equations”,J. Comput. Appl. Math., Vol.: 235,pp.:4871-4877, 2011.

XXXIII.O. A. Taiwo,”A parameter expansion method for two-point nonlinear singularly perturbed boundary value problems”,Int. J. Comput. Math., Vol.:55, Issue: 3-4, pp.: 189-196, 1995.

XXXIV.S. H. Dong,”The ansatz method for analyzing Schrodinger’s equation with three anharmonic potentials in D dimensions”,J. Genetic Counse., Vol.: 15, Issue: 4, pp.: 385-395, 2002.

XXXV.S. Kumar, K. Sing, R. K. Gupta,”Coupled Higgs field equations and Hamiltonian amplitude equation: Lie classical approach and (𝐺′/𝐺)-expansion method”,Prama. J. Phys., Vol.: 79, Issue: 1, pp.: 41-60, 2012.

XXXVI.S. Kutluay, A. Esen,”Exp function method for solving the general improved KdV equation”,Int. J. Nonlinear Sci. Numer. Simul., Vol.: 10, Issue: 6, pp.: 717-725, 2009

XXXVII.S. Sirisubtawee, S. koonprasert,”Exact traveling wave solution of certain nonlinearpartial differential equations using the (𝐺′𝐺2)-expansion method”,Advan. Math. Phys., Article ID 7628651, pp.:15, 2018.

XXXVIII.X. J. Yang, H. M. Srivastava, J. H. He, D. Baleanu,”Cantor-type cylindrical co-ordinate method for differential equations with local fractional derivatives”,Phys. Lett. A, Vol.: 377, Issue: 28-30, pp.: 1696-1700, 2013.

XXXIX.Y. C. Hon, E. G. Fan, “A series of the exact solution for coupled Higgs field equations and coupled Schrodinger-Boussinesq equations”, Nonlinear Anal., Theory Methods Appl., Vol.: 71, Issue: 7-8, pp.: 3501-3508, 2009.

XL.Z. Islam,M. M. Hossain, M. A. W. Seikh, “Exact traveling wave solution to Benney-Luke equation”,J. Bangladesh Math. Soc., Vol.: 37, pp.:1-14, 2017.

H. M. Shahadat Ali, M. A. Habib, M. Mamun Miah, M. Ali Akbar View Download