#### Authors:

Md. Dulal Hossain,Ummey Kulsum,Md. Khorshed Alam,M. Ali Akbar,#### DOI NO:

https://doi.org/10.26782/jmcms.2018.10.00005#### Keywords:

Exact traveling wave solutions,Jimbo-Miwa equation,Calogero-Bogoyavlenskii-Schiff equation,new generalized(G1/G) -expansion method,#### Abstract

In this article, we form the exact wave solutions of the Jimbo-Miwa equation and the Calogero-Bogoyavlenskii-Schiff equation by applying the new generalized (G'/G)-expansion method. We explained the new generalized (G'/G)-expansion method to look for more general traveling wave solutions of the above mentioned equations. The traveling wave solutions attained by this method are in terms of hyperbolic, trigonometric and rational functions. The graphical representation of the obtained solutions is kink soliton, singular kink soliton, singular soliton and singular periodic solution. This method is very significant for extracting exact solutions of NLEEs which habitually occur in mathematical physics, engineering sciences and applied mathematics.#### Refference:

I.Abazari, R. and Abazari, R. “Hyperbolic, trigonometric and rational function solutions of Hirota-Ramani equation via -expansion method”. Math. Probl. Eng., 11, 424801 (2011)

II.Akbar, M. A. Ali, N. H. M. and Zayed, E. M. E. “A generalized and improved -expansion method for nonlinear evolution equations”. Math. Probl. Eng., pp: 22, Article ID 459879 (2012)

III.Akter, J. and Akbar, M. A. “Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method”. Res. Phys., 5, 125-130 (2015)

IV.Ali A. Seadawy, A. R. and Lu, D. “Soliton solutions of the nonlinear Schrodinger equation with the dual power law nonlinearity and resonant nonlinear Schrodinger equationand their modulation instability analysis”. Int. J. Light Electron Opt., doi.org/10.1016/j.ij1eo.2017.07.016 (2017)

V.Bekir, A. Guner, O. Bhrawy, A. H. and Biswas, A. “Solving nonlinear fractional differential equations using exp-function and -expansion methods, Romanian”. J. Phys., 60, 360-378 (2015)

VI.Bhrawy, A. H. Abdelkawy, M. A. and Biswas, A. “Topological solitons and cnoidal waves to a few nonlinear wave equations in theoretical physics”. Indian J. Phys. 87(11), 1125-1131 (2013)

VII.Cole, J. D. “On a quasi-linear parabolic equation occurring in aerodynamics”. Quart. Appl. Math., 9, 225-236 (1951) VIII.Feng, J. Li, W. and Wan, Q. “Using -expansion method to seek traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation” Appl. Math. Comput., 217, 5860-5865 (2011)

IX.Gepreel, K. A. Nofal, T. A. and Alasmari, A. A. “Exact solutions for nonlinear integro-partial differential equations using the generalized Kudryashov method”. J. Egyptian Math. Soc. 25, 438-444 (2017)

X.Hafez, M. G. Alam, M. N. and Akbar, M. A. “Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system”. J. King Saud University-Sci., 27, 105-112 (2015) old 30

XI.Hopf, E. “The partial differential equation ” Commun. Pure. Appl. Math., 3, 201-230 (1950)

XII.Hossain, A. K. M. K.S. and Akbar, M.A. “Traveling wave solutions of nonlinear evolution equations via modified simple equation method”. Int. J. Appl. Math. Theor. Phys., 3, 20-25 (2017)

XIII.Hossain, A. K. M. K. S. Akbar, M. A. and Wazwaz, A. M. “Closed form solutions of complex wave equations via modified simple equation method”. Cogent Phys., 4, 1312751 (2017)

XIV.Hosseini, K. Ayati, Z. andAnsari, R. “New exact solutions of the Tzitzeica-type equations in nonlinear optics using the exp-function method”. J. Modern Opt.,doi.org/10.1080/09500340.2017.1407002 (2017)

XV.Huang, Q. M. Qao, Y. T. Jai, S. L. Wang, Y. L. and Deng, G. F. “Bilinear Backlund transformation, soliton and periodic wave solutions for a (3+1)-dimensional variable-coefficientgeneralized shallow water wave equation” Nonlinear Dyn, 87, 2529-2540 (2017)

XVI.Irendaoreji, “New exact traveling wave solutions for the Kawahara and modified Kawahara equations”. Chaos solitons Fract., 19, 147-150 (2004)

XVII.Kabir,M. M. “Exact traveling wave solutions for nonlinear elastic rod equation”. J. King Saud University-Sci. xxx , xxx-xxx (2017)

XVIII.Khan, K. and Akbar, M. A. “Traveling wave solutions of the (2 +1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method”. Ain Shams Eng. J., 5, 247-256 (2013)

XIX.Kumar, A. Dayal, R. “Tanh-coth scheme for traveling wave solutions for nonlinear wave interaction model”. J. Egyptian Math. Soc., 23, 282-285 (2015)

XX.`Liu,J. G. and He, Y. “New periodic solitary wave solutions for the (3+1)-dimensional generalized shallow water equation”. Nonlinear Dyn., 90, 363-369 (2017)

XXI.Liu,J.G. andHe, Y. Abundant lump and lump-kink solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation, Nonlinear Dyn., 92, ( 3) 1103-1108 (2018)

XXII.Lu, X. andLin, F. “Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order”. Comm. Non. Sci. Num. Simul. 32, 241-261 (2016)

XXIII.Liu, J. G. Tian, Y. and Zeng, Z. F. “New exact periodic solitary-wave solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in multi-temperature electron plasmas”. American Ins. Phys., 7, 105013 (2017)

XXIV.Liu,J. G. Zhou, L. and He, Y. “Multiple soliton solutions for the new dimensional Korteweg-de Vries equation by multiple exp-function method”. Appl. Math. Lett., 80, 71-78 (2018)

XXV.Malfliet, W. “Solitary wave solutions of nonlinear wave equations”. American J. Phys., 60, 650-654 (1992)

XXVI.Malwe, B. H. Betchewe, G. Doka, S. Y. and Kofane, T.C. “Travelling wave solutions and soliton solutions for the nonlinear transmission line using the generalized Riccati equation mapping method”. Nonlinear Dyn, 84, 171-177 (2015) XXVII.Naher, H. and Abdullah, F. A. “The basic -expansion method for the fourth order Boussinesq equation”. Appl. Math., 3(10), 1144-1152 (2012)

XXVIII.Naher, H. and Abdullah, F. A. “New approach of -expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation”. AIP Adv., 3(3), 032116 (2013)

XXIX.Naher, H. Abdullah, F. A. and Akbar, M. A. “The -expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation”. Math. Probl. Eng., 2011, 11 (2011)

XXX.Nofal, T.A. “Simple equation method for nonlinear partial differential equations and its applications”. J. Egyptian Math. Soc., 24, 204-209 (2016)

XXXI.Roshid, H. O. “Novel solitary wave solution in shallow water and ion acoustic plasma waves in-terms of two nonlinear models via MSE method”. J. Ocean Eng. Sci., 2, 196-202 (2017)

XXXII.Sonmezoglu, A. Yao, M. Ekici, M. Mirzazadeh, M. and Zhou, Q. “Explicit solitons in the parabolic law nonlinear negative-index materials”. Nonlinear Dyn., 88, 595-607 (2017)

XXXIII.Wang, M. Li, X. and Zhang, J. “The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics”. Phys. Lett. A, 372, 417-423 (2008)

XXXIV.Wazwaz, A. M. “Exact soliton and kink solutions for new (3+1)-dimensional nonlinear modified equations of wave propagation”. Open Eng. 7, 169-174 (2017)

XXXV.Wazwaz, A. M. The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput., 188, 1467-1475 (2007)

XXXVI.Zayed, E. M. E. and Al-Nowehy, A. G. “Exact solutions of the Biswas-Milovic equation, the ZK (m, n, k) equation and the K (m, n) equation using the generalized Kudryashov method”. Open Phys., 14, 129-139 (2016)

XXXVII.Zayed, E. M. E. and Al-Nowehy, A. G. “Exact traveling wave solutions for nonlinear PDEs in mathematical physics using the generalized Kudryashov method”. Serbian J. Electrical Eng., 13, 203-227 (2016)

XXXVIII.Zayed, E. M. E. and Al-Nowehy, A. G. “Solitonsand other exact solutions for variant nonlinear Boussinesq equations, Int. J. Light Electron Opt., doi.org/10.1016/j.ij1eo. 03.092 (2017)

XXXIX.Zayed, E. M. E. and El-Malky, M. A. “The extended -expansion method and its applications for solving the (3+1)-dimensional nonlinear evolution equations in mathematical physics”. Glob. J. Sci. Frontier Res., 11, 68-80 (2011) XL.Zayed, E. M. E. and Gepreel, K. A. The -expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics, J. Math. Phys., 50, 013502 (2009)

XLI.Zhang, J. Jiang, F. and Zhao, X. “An improved -expansion method for solving nonlinear evolution equations”. Int. J. Comput. Math., 87, 1716-1725 (2010)

Author(s): Md. Dulal Hossain, Ummey Kulsum, Md. Khorshed Alam M. Ali Akbar View Download