Exploration on traveling wave solutions to the 3rd-order klein–fock-gordon equation (KFGE) in mathematical physics


  • Nur Hasan Mahmud Shahen European University of Bangladesh
  • Foyjonnesa . European University of Bangladesh
  • Md. Habibul Bashar European University of Bangladesh






The Expansion Method, Nonlinear Evolution Equation, Exact Solution, 3rd-Order Klein–Gordon Equation, Mathematical Physics.


In this paper, the -expansion method has been applied to find the new exact traveling wave solutions of the nonlinear evaluation equations (NLEEs) by utilizing 3rd-order Klein–Gordon Equation (KFGE). With the collaboration of symbolic commercial software maple, the competence of this method for inventing these exact solutions has been more exhibited. As an upshot, some new exact solutions are obtained and signified by hyperbolic function solutions, different combinations of trigonometric function solutions, and exponential function solutions. Moreover, the -expansion method is a more efficient method for exploring essential nonlinear waves that enrich a variety of dynamic models that arises in nonlinear fields. All sketching is given out to show the properties of the innovative explicit analytic solutions. Our proposed method is directed, succinct, and reasonably good for the various nonlinear evaluation equations (NLEEs) related treatment and mathematical physics also.



[1] Nakamura, Akira. "Surface Impurity Localized Diode Vibration of the Toda Lattice: Perturbation Theory Based on Hirota’s Bilinear Transformation Method." Progress of Theoretical Physics 61.2 (1979): 427-442. https://doi.org/10.1143/PTP.61.427.

[2] Hietarinta, Jarmo. "A search for bilinear equations passing Hirota’s threeâ€soliton condition. I. KdVâ€type bilinear equations." Journal of Mathematical Physics 28.8 (1987): 1732-1742. https://doi.org/10.1063/1.527815.

[3] Khan K, Akbar MA. Application of the exp(-Φ(ξ))-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation. World Appl. Sci.J.2013;24(10):13731377.

[4] Zhao, Mm, Li,C.The exp(-Φ(ξ))-expansion method applied to nonlinear evolution equations; 2008.

[5] Roshid HO, Rahman MA. The exp(-Φ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results Phys; 2014. https://doi.org/10.1016/j.rinp.2014.07.006.

[6] He, Yinghui, Shaolin Li, and Yao Long. "Exact solutions of the Klein-Gordon equation by modified Exp-function method." Int. Math. Forum. Vol. 7. No. 4. 2012.

[7] Kumar, Dipankar, and Melike Kaplan. "New analytical solutions of (2+ 1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques." Chinese journal of physics 56.5 (2018): 2173-2185. https://doi.org/10.1016/j.cjph.2018.09.013.

[8] Khater, Mostafa M. A., and Emad HM Zahran. "New Method to Evaluating Exact and Traveling Wave Solutions for Non-Linear Evolution Equations." J. Comput. Theor. Nanosci 12 (2015): 1-9.

[9] Abdou, M. A. "The extended tanh method and its applications for solving nonlinear physical models." Applied mathematics and computation 190.1 (2007): 988-996. https://doi.org/10.1016/j.amc.2007.01.070.

[10] Fan, Engui. "Extended tanh-function method and its applications to nonlinear equations." Physics Letters A 277.4-5 (2000): 212-218. https://doi.org/10.1016/S0375-9601(00)00725-8.

[11] Wang, Gang-Wei, and Tian-Zhou Xu. "Group analysis and new explicit solutions of simplified modified Kawahara equation with variable coefficients." Abstract and Applied Analysis. Vol. 2013. Hindawi, 2013. https://doi.org/10.1155/2013/139160.

[12] Wang, Gang-Wei, et al. "Singular solitons, shock waves, and other solutions to potential KdV equation." Nonlinear Dynamics 76.2 (2014): 1059-1068. https://doi.org/10.1007/s11071-013-1189-9.

[13] Wang, Gang-Wei, et al. "Solitons and Lie group analysis to an extended quantum Zakharov–Kuznetsov equation." Astrophysics and Space Science 349.1 (2014): 317-327. https://doi.org/10.1007/s10509-013-1659-z.

[14] Wang, Gang-wei, Xi-qiang Liu, and Ying-yuan Zhang. "New explicit solutions of the fifth-order KdV equation with variable coefficients." Bull. Malays. Math. Sci. Soc 37.3 (2014): 769-778.

[15] Jawad, Anwar Ja’afar Mohamad, Marko D. Petković, and Anjan Biswas. "Modified simple equation method for nonlinear evolution equations." Applied Mathematics and Computation 217.2 (2010): 869-877. https://doi.org/10.1016/j.amc.2010.06.030.

[16] Zayed, Elsayed ME. "A note on the modified simple equation method applied to Sharma–Tasso–Olver equation." Applied Mathematics and Computation 218.7 (2011): 3962-3964. https://doi.org/10.1016/j.amc.2011.09.025.

[17] Bai, Cheng-Lin, and Hong Zhao. "Complex hyperbolic-function method and its applications to nonlinear equations." Physics Letters A 355.1 (2006): 32-38. https://doi.org/10.1016/j.physleta.2006.01.094.

[18] Wang, Mingliang, Xiangzheng Li, and Jinliang Zhang. "Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation." Physics Letters A 363.1-2 (2007): 96-101. https://doi.org/10.1016/j.physleta.2006.10.077.

[19] Wang, Deng-Shan, Yu-Jie Ren, and Hong-Qing Zhang. "Further extended sinh-cosh and sin-cos methods and new non traveling wave solutions of the (2+ 1)-dimensional dispersive long wave equations." Appl. Math. E-Notes 5 (2005): 157-163.

[20] Wang, Mingliang, Xiangzheng Li, and Jinliang Zhang. "The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics." Physics Letters A 372.4 (2008): 417-423. https://doi.org/10.1016/j.physleta.2007.07.051.

[21] Islam ME, Khan K, Akbar MA, Islam R. Traveling wave solutions of nonlinear evolution equation via enhanced (G'/G)-expansion method. GANIT J. Bangladesh Math. Soc. 2013; 33:83-92. https://doi.org/10.3329/ganit.v33i0.17662.

[22] Bekir, Ahmet. "Application of the (G′ G)-expansion method for nonlinear evolution equations." Physics Letters A 372.19 (2008): 3400-3406. https://doi.org/10.1016/j.physleta.2008.01.057.

[23] Islam MH, Khan K, Akbar MA, Salam MA. Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous burgers equation. Springer Plus.2014;3:105. https://doi.org/10.1186/2193-1801-3-105.

[24] Ali, Ahmad T. "New generalized Jacobi elliptic function rational expansion method." Journal of computational and applied mathematics 235.14 (2011): 4117-4127. https://doi.org/10.1016/j.cam.2011.03.002.

[25] Fan, Engui, and Hongqing Zhang. "A note on the homogeneous balance method." Physics Letters A 246.5 (1998): 403-406. https://doi.org/10.1016/S0375-9601(98)00547-7.

[26] Wang, Mingliang. "Exact solutions for a compound KdV-Burgers equation." Physics Letters A 213.5-6 (1996): 279-287. https://doi.org/10.1016/0375-9601(96)00103-X.

[27] Sharma, Dinkar, Prince Singh, and Shubha Chauhan. "Homotopy perturbation transform Method with He’s polynomial for solution of coupled nonlinear partial differential equations." Nonlinear Engineering 5.1 (2016): 17-23. https://doi.org/10.1515/nleng-2015-0029.

[28] Bayat, Mahmoud, Iman Pakar, and Ganji Domairry. "Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review." Latin American Journal of Solids and Structures 9.2 (2012): 1-93. https://doi.org/10.1590/S1679-78252012000200003.

[29] Zayed, E. M. E., and Khaled A. Gepreel. "A series of complexiton soliton solutions for nonlinear Jaulent—Miodek PDEs using the Riccati equations method." Proceedings of the Royal Society of Edinburgh Section A: Mathematics 141.5 (2011): 1001-1015. https://doi.org/10.1017/S0308210510000405.

[30] He, Ji-Huan. "Variational iteration method for autonomous ordinary differential systems." Applied Mathematics and Computation 114.2-3 (2000): 115-123. https://doi.org/10.1016/S0096-3003(99)00104-6.

[31] Akbulut, Arzu, and Filiz Taşcan. "Application of conservation theorem and modified extended tanh-function method to (1+ 1)-dimensional nonlinear coupled Klein–Gordon–Zakharov equation." Chaos, Solitons & Fractals 104 (2017): 33-40. https://doi.org/10.1016/j.chaos.2017.07.025.

[32] Hafez, M. G., Md Nur Alam, and M. Ali Akbar. "Exact traveling wave solutions to the Klein–Gordon equation using the novel (G′/G)-expansion method." Results in Physics 4 (2014): 177-184. https://doi.org/10.1016/j.rinp.2014.09.001.

View Full Article: