On new traveling wave solutions of potential KdV and (3+1)-dimensional Burgers equations

Volume 9, Issue 7, pp 5029--5040 http://dx.doi.org/10.22436/jnsa.009.07.07

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Authors

Fairouz Tchier - Department of Mathematics, King Saud University, P. O. Box 22452, Riyadh 11495, Saudi Arabia. Ibrahim E. Inan - Faculty of Education, Firat University, 23119 Elazig, Turkey. Yavuz Ugurlu - Science Faculty, Department of Mathematics, Firat University, 23119 Elazig, Turkey. Mustafa Inc - Science Faculty, Department of Mathematics, Firat University, 23119 Elazig, Turkey. Dumitru Baleanu - Department of Mathematics, Cankaya University, Ogretmenler Cad. 14 06530, Balgat, Ankara, Turkey. - Institute of Space Sciences, Magurele-Bucharest, Romania.

Abstract

This paper acquires soliton solutions of the potential KdV (PKdV) equation and the (3+1)-dimensional Burgers equation (BE) by the two variables \((\frac{G'}{ G} ,\frac{ 1}{ G})\) expansion method (EM). Obtained soliton solutions are designated in terms of kink, bell-shaped solitary wave, periodic and singular periodic wave solutions. These solutions may be useful and desirable to explain some nonlinear physical phenomena.

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Keywords

• \((\frac{G'}{ G}، \frac{ 1}{ G})\) -EM
• the PKdV equation
• the (3+1)-dimensional BE
• hyperbolic solution
• periodic solution
• rational solution.

MSC

•  35Q53
•  35C08
•  35C07

References

• [1] C. Q. Dai, Y. Y. Wang, New exact solutions of the (3+1)-dimensional Burgers system, Phys. Lett. A, 373 (2009), 181--187


• [2] C. Q. Dai, F. B. Yu, Special solitonic localized structures for the (3 + 1)-dimensional Burgers equation in water waves, Wave Motion, 51 (2014), 52--59


• [3] S. A. Elwakil, S. K. El-labany, M. A. Zahran, R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A, 299 (2002), 179--188


• [4] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212--218


• [5] Z. Fu, S. Liu, S. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 290 (2001), 72--76


• [6] S. Guo, Y. Zhou, The extended (\(\frac{G'}{ G}\) )-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations, Appl. Math. Comput., 215 (2010), 3214--3221


• [7] J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700--708


• [8] L. Li, E. Li, M. Wang, The (\(\frac{G'}{ G} ,\frac{ 1}{ G}\) )-expansion method and its application to travelling wave solutions of the Zakharov equations, Appl. Math. J. Chinese Univ. Ser. B, 25 (2010), 454--462


• [9] H. L. Lü, X. Q. Liu, L. Niu, A generalized (\(\frac{G'}{ G})\)-expansion method and its applications to nonlinear evolution equations, Appl. Math. Comput., 215 (2010), 3811--3816


• [10] B. Lu, H. Q. Zhang, F. D. Xie, Travelling wave solutions of nonlinear partial equations by using the first integral method, Appl. Math. Comput., 216 (2010), 1329--1336


• [11] W. Mal iet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys., 60 (1992), 650--654


• [12] E. J. Parkes, B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun., 98 (1996), 288--300


• [13] M. Wang, X. Li, J. Zhang, The (\(\frac{G'}{ G})\)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417--423


• [14] G. W. Wang, T. Z. Xu, G. Ebadi, S. Johnson, A. J. Strong, A. Biswas, Singular solitons, shock waves, and other solutions to potential KdV equation, Nonlinear Dynam., 76 (2014), 1059--1068


• [15] A. M. Wazwaz, Multiple soliton solutions and multiple singular soliton solutions for the (3 + 1)-dimensional Burgers equations, Appl. Math. Comput., 204 (2008), 942--948


• [16] E. M. E. Zayed, M. A. M. Abdelaziz, The two-variable (\(\frac{G'}{ G} ,\frac{ 1}{ G}\) )-expansion method for solving the nonlinear KdV-mKdV equation, Math. Probl. Eng., 2012 (2012), 14 pages


• [17] E. M. E. Zayed, K. A. E. Alurrfi, The (\(\frac{G'}{ G} ,\frac{ 1}{ G}\) )-expansion method and its applications for solving two higher order nonlinear evolution equations, Math. Probl. Eng., 2014 (2014), 20 pages


• [18] E. M. E. Zayed, S. A. Hoda Ibrahim, M. A. M. Abdelaziz, Traveling wave solutions of the nonlinear (3 + 1)- dimensional Kadomtsev-Petviashvili equation using the two variables (\(\frac{G'}{ G} ,\frac{ 1}{ G}\) )-expansion method, J. Appl. Math., 2012 (2012), 8 pages


• [19] X. Zheng, Y. Chen, H. Zhang, Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation, Phys. Lett. A, 311 (2003), 145--157