Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations

Year 2018, Volume: 1 Issue: 4, 273 - 279, 20.12.2018

İbrahim Enam İnan

https://doi.org/10.32323/ujma.399596

Cited By: 7

Abstract

In this paper, we implemented an improved tanh function Method for multiple soliton solutions of new coupled Konno-Oono equation and extended (3+1)-dimensional KdV-type equation.

Keywords

New coupled Konno-Oono equation , extended (3+1)-dimensional KdV-type equation , Improvedtanh function method , Multiple soliton solutions

References

• [1] L. Debtnath, Nonlinear partial differential equations for scientist and engineers, Birkhauser, Boston, MA, 1997.
• [2] A. M. Wazwaz, Partial differential equations: methods and applications, Balkema, Rotterdam, 2002.
• [3] Y. Shang, Backlund transformation,Lax pairs and explicit exact solutions for the shallow water wave sequation, Appl. Math. Comput., 187 (2007), 1286-1297.
• [4] T. L. Bock, M. D. Kruskal, A two-parameter Miura transformation of the Benjamin-Onoequation, Phys. Lett. A, 74 (1979), 173-176.
• [5] V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Springer, Berlin, 1991.
• [6] A.M. Abourabia, M. M. El Horbaty, On solitary wave solutions for the two-dimensional nonlinear modified Kortweg-de Vries-Burger equation, Chaos Solitons Fractals, 29 (2006), 354-364.
• [7] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.
• [8] Y. Chuntao, A simple transformation for nonlinear waves, Phys. Lett. A, 224 (1996), 77-84.
• [9] F. Cariello, M. Tabor, Painlev eexpansions for nonintegrable evolution equations, Phys. D, 39(1989), 77-94.
• [10] E. Fan, Two new application of the homogeneous balance method, Phys. Lett. A, 265 (2000), 353-357.
• [11] P. A. Clarkson, New similarity solutions for the modified boussinesq equation, J. Phys. A: Math. Gen., 22 (1989), 2355-2367.
• [12] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.
• [13] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218.
• [14] 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.
• [15] H. Chen, H. Zhang, New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation, Chaos Solitons Fractals, 19 (2004), 71-76.
• [16] Z. Fu, S. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 290 (2001), 72-76.
• [17] S. Shen, Z. Pan, A note on the Jacobi elliptic function expansion method, Phys. Let. A, 308 (2003), 143-148.
• [18] H. T. Chen, Z. Hong-Qing, New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation, Chaos Solitons Fractals, 20 (2004), 765-769.
• [19] Y. Chen, Q. Wang, B. Li, Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly periodic solutions of nonlinear evolution equations, Z. Naturforsch. A, 59 (2004), 529-536.
• [20] Y. Chen, Z. Yan, The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos Solitons Fractals, 29 (2006), 948-964.
• [21] M. Wang, X. Li, J. Zhang, The $\left( {\frac{{G'}}{G}} \right)$-expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics, Phys. Lett. A, 372 (2008), 417-423.
• [22] S. Guo, Y. Zhou, The extended $\left( {\frac{{G'}}{G}} \right)$-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.
• [23] H. L. Lu, X. Q. Liu, L. Niu, A generalized $\left( {\frac{{G'}}{G}} \right)$-expansion method and its applications to nonlinear evolution equations, Appl. Math. Comput., 215 (2010), 3811-3816.
• [24] L. Li, E. Li, M. Wang, The $\left( {\frac{{G'}}{G},\frac{1}{G}} \right)$ -expansion method and its application to travelling wave solutions of the Zakharov equations, Appl. Math-A J. Chin. U., 25 (2010), 454-462.
• [25] J. Manafian, Optical soliton solutions for Schrodinger type nonlinear evolution equations by the tan $\left( {\frac{{\phi \left( \varphi \right)}}{2}} \right)$- expansion Method, Optik, 127 (2016), 4222-4245.
• [26] E. Don, Schaum’s outline of Theoryand problems of mathematica, McGraw-Hill, 2001.
• [27] E. Yasar, New travelling wave solutions to the Ostrovsky equation, Appl. Math. Comput., 216(11) (2010), 3191-3194.
• [28] Y. Yildirim, E. Yasar, An extended Korteweg-de Vries equation: multi-soliton solutions and conservation laws, Nonlinear Dynam., 90(3) (2017), 1571-1579.
• [29] E. Yasar, S. San, Y. S. Ozkan, Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation, Open Phys., 14(1) (2016), 37-43.
• [30] W. X. Ma, Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differential Equations, 264(4) (2018), 2633-2659.
• [31] W. X. Ma, T. Huang, Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Physica Scr., 82(6) (2010), 065003.
• [32] M. S. Osman, H. I. Abdel-Gawad, M. A. El Mahdy, Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion, Results Phys., 8 (2018), 1054-1060.
• [33] Hamdy I. Abdel-Gawad, M. S. Osman, On the variational approach for analyzing the stability of solutions of evolution equations, Kyungpook Math., 53 (2013), 661-680.
• [34] M. S. Osman, On complex wave solutions governed by the 2D Ginzburg-Landau equation with variable coefficients, Optik, 156 (2018), 169-174.
• [35] M. S. Osman, J. A. T. Machado, D. Baleanu, On nonautonomous complex wave solutions described by the coupled Schrodinger-Boussinesq equation with variable-coefficients, Opt. Quant. Electron, 50 (2018), 73.
• [36] M. Nasir Ali, M. S. Osman, S. Muhammad Husnine, On the analytical solutions of conformable time-fractional extended Zakharov-Kuznetsov equation through $\left( {\frac{{G'}}{{G2}}} \right)$-expansion method and the modified Kudryashov method, SeMA Journal, (in Press).
• [37] M. S. Osman, Multiwave solutions of time-fractional (2 + 1)-dimensional Nizhnik-Novikov-Veselov equations, Pramana J. Phys., 88 (2017), 67.
• [38] M. S. Osman, A. Majid Wazwaz, An efficient algorithm to construct multi-soliton rational solutions of the (2+ 1)-dimensional KdV equation with variable coefficients, Appl. Math. Comput., 321 (2018), 282-289.
• [39] M. S. Osman, On multi-soliton solutions for the (2 + 1)-dimensional breaking soliton equation with variable coefficients in a graded-index waveguide, Comput. Math. Appl., 75 (2018), 1-6.